%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: wcs.dvi %%Pages: 42 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips wcs %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1996.09.09:0941 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 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y(of)j(its)h(commissions,)d(including)i(the)h(FITS)g(resolution.)1032 868 y(The)g(original)e(pap)q(er)i(an)o(ticipated)f(the)h(need)g(to)f (sp)q(ecify)h(the)970 918 y(co)q(ordinates)k(to)e(b)q(e)h(attac)o(hed)h (to)e(eac)o(h)i(pixel.)d(It)i(view)o(ed)g(eac)o(h)970 968 y(axis)i(of)f(the)i Fr(n)p Fw(-dimensional)d(image)g(as)i(ha)o (ving)e(a)i(co)q(ordinate)970 1017 y(t)o(yp)q(e)14 b(and)e(a)h (reference)j(p)q(oin)o(t)c(for)h(whic)o(h)g(the)g(pixel)g(co)q (ordinate,)970 1067 y(a)c(co)q(ordinate)h(v)n(alue,)e(and)h(an)g (incremen)o(t)g(w)o(ere)h(giv)o(en.)f(Note)g(that)970 1117 y(this)17 b(reference)h(p)q(oin)o(t)e(w)o(as)h(not)f(required)h (to)f(o)q(ccur)i(at)e(in)o(teger)970 1167 y(pixel)j(lo)q(cations)f(nor) i(ev)o(en)f(to)g(o)q(ccur)i(within)d(the)i(image.)c(An)970 1217 y(unde\014ned)c(\\rotation")d(parameter)g(w)o(as)i(also)e(pro)o (vided)h(for)g(eac)o(h)970 1266 y(axis.)16 b(Since)i(there)g(are,)f(in) f(general,)h(more)f(co)q(ordinates)h(to)g(b)q(e)970 1316 y(attac)o(hed)h(to)g(a)f(pixel)g(than)h(there)h(are)f(\\real")f(axes)h (in)f(the)h Fr(n)p Fw(-)970 1366 y(dimensional)12 b(image,)g(the)j(con) o(v)o(en)o(tion)f(of)g(declaring)g(axes)h(with)970 1416 y(a)e(single)g(pixel)f(w)o(as)h(also)g(established)h(in)f(b)q(oth)g (examples)f(giv)o(en)970 1466 y(b)o(y)i(W)m(ells,)e Fr(et)j(al)p Fw(.)e(The)i(k)o(eyw)o(ords)f(de\014ned)h(w)o(ere)977 1537 y Fp(CRVAL)p Fr(n)49 b Fw(co)q(ordinate)14 b(v)n(alue)f(at)h (reference)i(p)q(oin)o(t)977 1587 y Fp(CRPIX)p Fr(n)49 b Fw(arra)o(y)13 b(lo)q(cation)g(of)h(the)g(reference)i(p)q(oin)o(t)e (in)f(pixels)977 1637 y Fp(CDELT)p Fr(n)49 b Fw(co)q(ordinate)14 b(incremen)o(t)g(at)f(reference)k(p)q(oin)o(t)977 1686 y Fp(CTYPE)p Fr(n)49 b Fw(axis)13 b(t)o(yp)q(e)i(\(8)e(c)o(haracters\)) 977 1736 y Fp(CROTA)p Fr(n)49 b Fw(rotation)13 b(from)f(stated)j(co)q (ordinate)f(t)o(yp)q(e)970 1808 y(A)j(list)f(of)g(suggested)j(v)n (alues)d(for)h Fp(CTYPE)p Fr(n)f Fw(w)o(as)g(pro)o(vided)h(with)970 1858 y(few)12 b(of)g(the)h(details)f(actually)f(required)i(to)f(sp)q (ecify)h(co)q(ordinates.)970 1908 y(The)j(units)g(w)o(ere)h(sp)q (eci\014ed)g(to)f(b)q(e)g(The)g(In)o(ternational)f(System)970 1958 y(of)f(Units)i(\\SI")e(\(meters,)h(grams,)e(seconds\))k(with)d (the)i(addition)970 2008 y(of)d(degrees)j(for)e(angles.)1032 2057 y(The)k(simplicit)o(y)d(of)i(this)g(initial)f(description)i(w)o (as)f(delib)q(er-)970 2107 y(ate.)e(It)f(w)o(as)h(felt)f(that)g(a)h (detailed)f(sp)q(eci\014cation)h(of)f(co)q(ordinate)970 2157 y(t)o(yp)q(es)d(w)o(as)f(a)f(length)o(y)h(and)g(complicated)e (business,)j(w)o(ell)e(b)q(ey)o(ond)970 2207 y(the)17 b(scop)q(e)g(in)o(tended)f(for)g(the)h(initial)c(pap)q(er.)k(In)f (addition,)e(the)970 2257 y(authors)j(felt)f(that)g(a)g(detailed)h(sp)q (eci\014cation)g(w)o(ould)e(probably)970 2307 y(b)q(e)g(somewhat)d(con) o(tro)o(v)o(ersial)i(and)f(th)o(us)i(lik)o(ely)d(to)i(compromise)970 2356 y(the)k(p)q(ossibilit)o(y)e(of)h(wide-spread)h(agreemen)o(t)f(on,) g(and)g(use)i(of,)970 2406 y(the)13 b(basic)e(structures)k(of)c(the)h (format.)e(Hindsigh)o(t)h(also)g(suggests)970 2456 y(that)19 b(w)o(e)g(w)o(ere)h(rather)f(naiv)o(e)f(at)h(the)g(time)f(concerning)h (co)q(or-)970 2506 y(dinates)d(and)f(it)f(is)h(fortunate)h(that)f(the)h (detailed)f(sp)q(eci\014cation)970 2556 y(w)o(as)c(p)q(ostp)q(oned)h (un)o(til)f(greater)h(exp)q(erience)h(could)e(b)q(e)h(obtained.)1032 2605 y(While)18 b(participating)g(in)g(the)h(dev)o(elopmen)o(t)e(of)h (the)h(AIPS)970 2655 y(soft)o(w)o(are)f(pac)o(k)n(age)g(of)g(the)h (National)e(Radio)f(Astronom)o(y)h(Ob-)p eop %%Page: 2 2 2 1 bop -116 -192 a Fs(2)449 b(Greisen)14 b(&)e(Calabretta:)i(Represen) o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 -93 y Fw(serv)n(atory)m(,)h(Greisen)h(\(1983)f(and)h(1986)o(\))g(found) f(it)g(necessary)j(to)-116 -43 y(supply)c(additional)e(details)i(to)g (the)h(co)q(ordinate)f(de\014nitions)g(for)-116 7 y(b)q(oth)19 b(v)o(elo)q(cit)o(y)g(and)g(celestial)g(co)q(ordinates.)h(The)g(latter) f(ha)o(v)o(e)-116 57 y(b)q(een)d(widely)f(used)h(for)f(imagery)e(from)h (the)i(VLA)f(and)h(a)f(n)o(um-)-116 106 y(b)q(er)g(of)g(other)g (instrumen)o(ts)g(\(Moshir,)g Fr(et.)h(al)p Fw(,)e(1992)g(for)h (IRAS\).)-116 156 y(Because)21 b(of)e(this)h(wide)f(usage,)h(w)o(e)g (require)g(that)g(an)o(y)f(future)-116 206 y(sp)q(eci\014cation)14 b(supp)q(ort)h(the)f(AIPS)g(description)h(of)e(co)q(ordinates)-116 256 y(wherev)o(er)e(p)q(ossible.)f(Greisen)g(\(1983\))g(assumed)f(that) h(there)h(is)f(no)-116 306 y(sk)o(ew)k(in)g(the)g(image)e(co)q (ordinates)j(and)f(that)g(the)g(only)g(rotation)-116 356 y(applied)g(to)h(the)h(celestial)f(co)q(ordinate)g(pair.)f(F)m(or)h (that)g(pair,)f(he)-116 405 y(in)o(tro)q(duced)i(left-handed,)f(linear) g(co)q(ordinates)h(in)f(the)h(tangen)o(t)-116 455 y(plane,)d(whic)o(h)h (are)g(giv)o(en)f(b)o(y)-116 543 y Fo(x)20 b Fw(=)h(\()p Fo(i)10 b Fn(\000)f Fp(CRPIX)p Fr(i)o Fw(\))e Fp(CDELT)p Fr(i)-116 593 y Fo(y)24 b Fw(=)d(\()p Fo(j)12 b Fn(\000)d Fp(CRPIX)p Fr(j)o Fw(\))e Fp(CDELT)p Fr(j)f Fo(;)870 569 y Fw(\(1\))-116 682 y(where)13 b Fo(i;)7 b(j)15 b Fw(are)e(pixel)f(n)o(um)o(b)q(ers)g(in)g(the)h(relev)n(an)o(t)g(t)o(w)o (o)f(axes.)g(The)-116 732 y(pixel)j(n)o(um)o(b)q(ers)g(are)h(coun)o (ted)g(from)e(one)h(through)h Fp(NAXIS)p Fr(i)e Fw(and)-116 782 y Fp(NAXIS)p Fr(j)p Fw(,)d(resp)q(ectiv)o(ely)m(,)i(with)g(the)h (lo)q(cation)e(of)h(eac)o(h)g(pixel)g(tak)o(en)-116 832 y(to)c(b)q(e)h(its)g(cen)o(ter.)g(Therefore,)h(the)f(actual)f(image)f (area)h(runs)i(from)-116 882 y(\()p Fo(i;)c(j)r Fw(\))14 b(=)g(\(0)p Fo(:)p Fw(5)p Fo(;)7 b Fw(0)p Fo(:)p Fw(5\))13 b(to)j(\()p Fo(i;)7 b(j)r Fw(\))14 b(=)g(\()p Fp(NAXIS)p Fo(i)c Fw(+)h(0)p Fo(:)p Fw(5)p Fo(;)c Fp(NAXIS)m Fo(j)13 b Fw(+)d(0)p Fo(:)p Fw(5\).)-116 931 y(The)k(direction)g(cosines,)g(\() p Fo(L;)7 b(M)e Fw(\))14 b(are)g(then)h(giv)o(en)e(b)o(y)-116 1022 y Fo(L)37 b Fw(=)21 b Fo(x)7 b Fw(cos)g Fo(\032)j Fn(\000)f Fo(y)g Fw(sin)e Fo(\032)-116 1072 y(M)25 b Fw(=)c Fo(y)9 b Fw(cos)e Fo(\032)j Fw(+)g Fo(x)d Fw(sin)f Fo(\032)h(;)870 1047 y Fw(\(2\))-116 1161 y(where)k Fo(L)g Fw(is)f(in)g(the)h(direction)f(of)g(constan)o(t)h(latitude)f(and)g Fo(M)15 b Fw(is)10 b(in)-116 1211 y(the)f(direction)h(of)f(constan)o(t) g(longitude.)g(The)g(rotation)g Fo(\032)h Fw(is)f(giv)o(en,)-116 1260 y(b)o(y)14 b(con)o(v)o(en)o(tion,)g(in)h(the)g Fp(CROTA)p Fr(n)g Fw(parameter)f(for)h(the)g(latitude-)-116 1310 y(lik)o(e)f(axis.)g(Note)i(that)f(these)i(equations)e(de\014ne)h Fo(\032)f Fw(as)h(the)f(angle)-116 1360 y(from)f(the)i(+)p Fo(y)i Fw(axis)d(to)g(the)i(+)p Fo(M)j Fw(axis)c(in)f(the)h(direction)g (of)f(the)-116 1410 y(+)p Fo(x)20 b Fw(axis,)f Fr(i.e.)p Fw(,)h(coun)o(ter-clo)q(c)o(kwise)h(for)f(left-handed)h(systems)-116 1460 y(and)13 b(clo)q(c)o(kwise)i(for)e(righ)o(t-handed)h(systems.)-54 1510 y(The)c(t)o(yp)q(e)g(of)f(longitude-latitude)g(w)o(as)g(sp)q (eci\014ed)i(b)o(y)f(the)g(\014rst)-116 1559 y(four)h(c)o(haracters)i (of)e Fp(CTYPE)p Fr(n)g Fw(as)h Fp('RA--')e Fw(and)h Fp('DEC-')g 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1169 y Fw(=)64 1048 y Fl(0)64 1121 y(B)64 1146 y(B)64 1171 y(B)64 1197 y(@)100 1080 y Fp(CDELT1)74 b(0)130 b(0)75 b Fo(:)7 b(:)g(:)155 1129 y Fp(0)75 b(CDELT2)f(0)h Fo(:)7 b(:)g(:)155 1179 y Fp(0)129 b(0)75 b(CDELT3)20 b Fo(:)7 b(:)g(:)160 1223 y Fw(.)160 1240 y(.)160 1257 y(.)312 1223 y(.)312 1240 y(.)312 1257 y(.)463 1223 y(.)463 1240 y(.)463 1257 y(.)557 1228 y(.)573 1240 y(.)590 1253 y(.)614 1048 y Fl(1)614 1121 y(C)614 1146 y(C)614 1171 y(C)614 1197 y(A)659 1169 y Fn(\002)64 1291 y Fl(0)64 1364 y(B)64 1389 y(B)64 1414 y(B)64 1441 y(@)100 1323 y Fp(PC001001)20 b(PC001002)f(PC001003)g Fo(:)7 b(:)g(:)100 1373 y Fp(PC002001)20 b(PC002002)f(PC002003)g Fo(:)7 b(:)g(:)100 1423 y Fp(PC003001)20 b(PC003002)f(PC003003)g Fo(:)7 b(:)g(:)182 1467 y Fw(.)182 1484 y(.)182 1500 y(.)377 1467 y(.)377 1484 y(.)377 1500 y(.)572 1467 y(.)572 1484 y(.)572 1500 y(.)688 1471 y(.)704 1484 y(.)720 1496 y(.)745 1291 y Fl(1)745 1364 y(C)745 1389 y(C)745 1414 y(C)745 1441 y(A)790 1412 y Fn(\002)48 b 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y(to)i(transp)q(ose) h(the)f(header)h(parameters)f(along)f(with)g(the)h(image)970 1414 y(so)i(that)f(the)h(on-diagonal)d(terms)j(in)e(the)j Fp(PC)d Fw(matrix)g(predomi-)970 1464 y(nate.)h(F)m(urthermore,)f(w)o (e)h(recommend)f(strongly)g(that)h(the)g Fr(sign)970 1514 y Fw(of)e(the)h(co)q(ordinate)f(incremen)o(t)g(b)q(e)h(represen)o (ted)i(in)c(the)i Fp(CDELT)p Fr(n)970 1563 y Fw(k)o(eyw)o(ords)f (rather)g(than)f(in)f(the)i Fp(PC)f Fw(matrix.)d(If)j(the)h Fp(PC)e Fw(matrix)g(is)970 1613 y(essen)o(tially)f(diagonal)d(and)j(p)q (ositiv)o(e,)e(then)i(the)h(h)o(uman)c(reader)k(of)970 1663 y(the)j(FITS)f(header)h(will)e(ha)o(v)o(e)h(a)g(b)q(etter)i(c)o (hance)f(of)e(understand-)970 1713 y(ing)h(the)h(co)q(ordinate)g (represen)o(tation.)g(Go)q(o)q(d)f(form)f(in)h(FITS)g(al-)970 1763 y(w)o(a)o(ys)f(extends)h(to)e(include)h(the)g(h)o(uman)e(as)i(w)o (ell)f(as)h(the)g(soft)o(w)o(are)970 1813 y(readers.)1032 1867 y(Tw)o(o)j(other)h(forms)e(of)g(the)i Fp(PC)p Fr(nnnmmm)g Fw(k)o(eyw)o(ord,)f(lab)q(eled)970 1916 y Fp(CD)p Fr(n)p 1040 1916 14 2 v 16 w(m)h Fw(and)g Fp(CD)p Fr(nnnmmm)p Fw(,)g(ha)o(v)o(e)f(also)h(b)q(een)h(used)g(and)f(should)970 1966 y(b)q(e)j(recognized)g(as)f(the)g(pro)q(duct)h(of)f(the)g Fp(PC)p Fr(nnnmmm)g Fw(matrix)p 970 2018 237 2 v 970 2046 a Ft(1)1028 2062 y Fs(The)9 b(authors)i(are)f(w)o(ell)h(a)o(w)o (are)e(that)h(this)h(con)o(v)o(en)o(tion)g(di\013ers)h(from)970 2107 y(the)k(usual)h(practice)g(in)f(computer)h(graphics)g(where)f(the) g(pixels)h(are)970 2153 y(coun)o(ted)k(from)f(zero)h(with)f(pixel)i (cen)o(ters)f(as)f(half)h(in)o(tegers)g(\()p Fq(e.g.)p Fs(,)970 2199 y(Adob)q(e)12 b(Systems,)f(Inc.,)f(1990)q(\).)g(The)h (con)o(v)o(en)o(tion)i(prop)q(osed)g(here)e(has)970 2244 y(b)q(een)16 b(used)g(extensiv)o(ely)h(in)f(FITS)f(since)i(the)e (format)g(w)o(as)g(in)o(v)o(en)o(ted)970 2290 y(and)e(no)g(argumen)o(t) g(has)f(b)q(een)h(adv)n(anced)h(su\016cien)o(tly)g(comp)q(elling)h(to) 970 2336 y(in)o(v)n(alidate)g(the)d(man)o(y)h(thousands)h(of)d(tap)q (es)i(written)f(with)h(that)f(con-)970 2381 y(v)o(en)o(tion.)f(F)m (urthermore,)f(w)o(e)g(regard)g(our)h(image)f(samples)i(as)e(\\v)o(o)o (xels")970 2427 y(in)g(real)h(ph)o(ysical)h(space)e(rather)f(than)i (pixels)g(in)f(t)o(w)o(o-dimensional)j(dis-)970 2473 y(pla)o(y)k(space.)f(As)f(suc)o(h,)h(they)g(ma)o(y)f(b)q(e)h(view)o(ed) g(from)g(an)o(y)g(angle)g(via)970 2518 y(transp)q(osition)21 b(and)e(rotation.)g(The)f(only)h(p)q(oin)o(t)h(within)f(the)f(v)o(o)o (xel)970 2564 y(that)12 b(remains)h(in)o(v)n(arian)o(t)g(under)g(those) f(op)q(erations)i(is)e(its)g(cen)o(ter)g(and)970 2610 y(w)o(e)g(argue,)h(therefore,)f(that)h(it)g(is)g(the)f(cen)o(ter)h(of)f (the)h(v)o(o)o(xel)g(whic)o(h)h(w)o(e)970 2655 y(should)h(coun)o(t.)p eop %%Page: 4 4 4 3 bop -116 -192 a Fs(4)449 b(Greisen)14 b(&)e(Calabretta:)i(Represen) o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 -93 y Fw(and)j(the)h Fp(CDELT)p Fr(n)f Fw(diagonal)f(matrix.)f(These)k (div)o(ergen)o(t)f(forms)-116 -43 y(should)13 b(no)h(longer)g(b)q(e)g (written.)-54 12 y(The)e(fourth)g(\(third)h(required\))g(step)g(in)f (the)g(pro)q(cess)i(of)e(\014nd-)-116 62 y(ing)f(the)h(true)h(co)q (ordinates)g(dep)q(ends)h(on)d(the)i(t)o(yp)q(e)f(of)g(axis)f(giv)o(en) -116 112 y(in)k Fp(CTYPE)p Fr(n)p Fw(.)g(F)m(or)h(simple)e(linear)i (axes,)g(the)h(true)g(co)q(ordinate)f(is)-116 162 y(found)d(b)o(y)h (adding)f(the)i(o\013set)g(found)e(ab)q(o)o(v)o(e)h(to)g(the)h(co)q (ordinate)-116 211 y(v)n(alue)f(at)h(the)h(reference)h(p)q(oin)o(t)e (giv)o(en)f(b)o(y)h Fp(CRVAL)p Fr(n)p Fw(.)f(Otherwise,)-116 261 y(some)g(function)g(of)h(the)g(o\013set\(s\),)h(the)g Fp(CRVAL)p Fr(n)p Fw(,)e(and,)g(p)q(erhaps,)-116 311 y(other)i(parameters)f(m)o(ust)g(b)q(e)h(established)g(b)o(y)g(con)o(v) o(en)o(tion)f(and)-116 361 y(agreemen)o(t.)h(Greisen)i(\(1983)o(\))f (giv)o(es)g(examples)f(of)g(the)i(consid-)-116 411 y(erations)c(whic)o (h)h(migh)o(t)d(apply)i(to)g(de\014ning)h(astronomical)d(\\v)o(e-)-116 461 y(lo)q(cit)o(y")g(axes.)-54 516 y(In)i(the)h(presen)o(t)g(man)o (uscript,)e(w)o(e)h(will)f(treat)i(only)e(the)i(con-)-116 566 y(v)o(en)o(tions)k(to)g(b)q(e)h(applied)f(to)g(pairs)g(of)g (astronomical)e(angular)-116 615 y(co)q(ordinates,)12 b Fr(i.e.)p Fw(,)e(to)i(celestial)g(longitudes)g(and)f(latitudes)h (with)-116 665 y(resp)q(ect)19 b(to)f(v)n(arious)f(systems)h(and)f (represen)o(ted)j(with)e(v)n(arious)-116 715 y(spherical)13 b(pro)r(jections.)h(F)m(or)f(this)g(case,)h(the)g(fourth)f(step)h(in)f (the)-116 765 y(computation)g(of)h(co)q(ordinate)i(v)n(alues)f(is)g(to) g(con)o(v)o(ert)h(the)f(linear)-116 815 y(\()p Fo(x;)7 b(y)q Fw(\))13 b(o\013sets)i(in)o(to)e(longitudes)g(and)g(latitudes)h (in)f(the)h(\\nativ)o(e")-116 864 y(co)q(ordinate)j(system)g(of)f(the)i (particular)f(spherical)g(pro)r(jection.)-116 914 y(The)c(t)o(yp)q(e)g (of)g(spherical)g(pro)r(jection)g(will)f(b)q(e)i(sp)q(eci\014ed)g(b)o (y)f(c)o(har-)-116 964 y(acters)19 b(5)e(through)h(8)f(of)g(the)i Fp(CTYPE)p Fr(n)d Fw(k)o(eyw)o(ord)i(and)g(m)o(ust)e(b)q(e)-116 1014 y(the)h(same)f(for)h(b)q(oth)g(axes)h(of)e(the)i(co)q(ordinate)f (pair.)g(The)g(par-)-116 1064 y(ticular)h(c)o(haracters)j(to)e(b)q(e)g (used)h(will)d(b)q(e)j(established)g(b)o(y)e(the)-116 1113 y(con)o(v)o(en)o(tions)e(describ)q(ed)i(in)e(Sections)d(4)j(and)g (7.)g(An)o(y)g(t)o(yp)q(e)h(not)-116 1163 y(so)e(sp)q(eci\014ed)i (should)e(b)q(e)h(tak)o(en)g(to)f(indicate)g(that)h(the)g(con)o(v)o (en-)-116 1213 y(tions)d(of)g(this)h(man)o(uscript)f(are)h(not)g(b)q (eing)g(used.)g(The)g(form)o(ul\032)-116 1263 y(for)19 b(the)i(con)o(v)o(ersion)f(from)e(\()p Fo(x;)7 b(y)q Fw(\))20 b(to)g(\()p Fo(\036;)7 b(\022)q Fw(\))20 b(will)e(also)h(b)q (e)i(de-)-116 1313 y(scrib)q(ed)14 b(in)e(Section)h(4.)f(The)i(nature)f (of)g(the)g(\\nativ)o(e")f(system)h(is)-116 1363 y(also)i(determined)h (en)o(tirely)h(b)o(y)f(the)h(pro)r(jection.)f(F)m(or)g(example,)-116 1412 y(for)g(zenithal)g(pro)r(jections,)h(the)g(nativ)o(e)f(system)g (has)h(its)g(North)-116 1462 y(p)q(ole)c(at)h(the)h(tangen)o(t)f(p)q (oin)o(t.)f(Some)g(pro)r(jections)i(require)g(addi-)-116 1512 y(tional)g(parametric)i(information.)c(These)19 b(parameters)e(will)e(b)q(e)-116 1562 y(giv)o(en)f(with)g(FITS)g(k)o (eyw)o(ords)h Fp(PROJP)p Fr(m)p Fw(,)e(for)h Fo(m)f Fw(=)g(0)p Fo(;)7 b Fw(1)p Fo(;)g Fw(2)p Fo(;)g(:)g(:)g(:)k Fw(as)-116 1612 y(sp)q(eci\014ed)k(in)e(the)i(follo)o(wing)c(descriptions)k(of)e (the)i(pro)r(jections.)-54 1667 y(The)21 b(\014fth)f(\(fourth)h (required\))g(step)h(in)e(the)h(computation)-116 1717 y(of)f(the)i(co)q(ordinate)g(v)n(alue)f(of)g(the)h(angle)f(pair)g (consists)h(of)f(a)-116 1766 y(spherical)g(co)q(ordinate)f(rotation)g (from)f(the)i(nativ)o(e)g(longitude)-116 1816 y(and)16 b(latitude)g(to)h(a)f(recognized)i(celestial)f(co)q(ordinate)g(system.) -116 1866 y(The)f(t)o(yp)q(e)g(of)f(the)h(standard)g(system)g(is)f (indicated)h(in)f(the)h(\014rst)-116 1916 y(four)f(c)o(haracters)j(of)e (the)g Fp(CTYPE)p Fr(n)g Fw(as)g Fp('RA--')f Fw(and)g Fp('DEC-')g Fw(for)-116 1966 y(equatorial)c(co)q(ordinates,)i Fp(')p Fo(x)p Fp(LON')f Fw(and)g Fp(')p Fo(x)p Fp(LAT')f Fw(for)i(longitude-)-116 2015 y(latitude)f(systems,)h(where)i Fo(x)c Fw(=)i Fp(G)g Fw(for)g(Galactic,)f Fp(E)h Fw(for)f(Ecliptic,) -116 2065 y Fp(H)18 b Fw(for)g(helio)q(ecliptic,)f(and)i Fp(S)f Fw(for)g(sup)q(ergalactic)h(co)q(ordinates.)904 2050 y Fu(2)-116 2115 y Fw(Other)12 b(v)n(alues)g(of)f Fo(x)g Fw(will)f(undoubtedly)i(b)q(e)g(de\014ned;)g(FITS)g(read-)-116 2165 y(ers)17 b(should)g(note)g(the)g(general)g(form)e(and)i(handle)g (the)g(co)q(ordi-)-116 2215 y(nate)10 b(pair)g(ev)o(en)g(if)f(the)i (particular)f(v)n(alue)f(of)h Fo(x)f Fw(is)h(not)g(recognized.)-116 2265 y(The)16 b Fp(CRVAL)p Fr(n)e Fw(giv)o(e)h(the)i(co)q(ordinates)f (in)f(the)h(standard)g(system)-116 2314 y(of)d(the)i(reference)i(p)q (oin)o(t)d(in)g(the)h(nativ)o(e)f(system)g(\()p Fr(e.g.)p Fw(,)g(the)h(tan-)-116 2364 y(gen)o(t)j(p)q(oin)o(t\).)f(Exactly)h (what)g(the)h(reference)i(p)q(oin)o(t)d(is)g(will)e(b)q(e)-116 2414 y(de\014ned)e(for)g(eac)o(h)g(of)f(the)h(pro)r(jections)h(describ) q(ed)g(in)f(Section)f(4.)-116 2464 y(A)j(full)f(co)q(ordinate)i (rotation)f(requires)h(a)f(third)h(angle)f(as)g(w)o(ell.)p -116 2520 237 2 v -116 2548 a Ft(2)-59 2564 y Fs(\\New")d(Galactic)h (co)q(ordinates)h(are)d(assumed)i(here,)f(1960)q(.)f(Users)-116 2610 y(of)g(the)i(older)g(system)f(or)g(future)h(systems)f(should)i (adopt)f(a)f(di\013eren)o(t)-116 2655 y(v)n(alue)h(of)f Fk(x)f Fs(and)i(do)q(cumen)o(t)g(its)f(meaning.)970 -93 y Fw(This)g(third)g(angle)f(w)o(as)h(a)o(v)o(oided)f(in)g(AIPS)h(b)o(y) g(Greisen's)g(\(1983)o(\))970 -43 y(requiremen)o(t)j(that)f(the)h (nativ)o(e)f(system)g(b)q(e)h(lo)q(cally)e(parallel)h(to)970 7 y(the)21 b(standard)f(system.)f(The)h(third)g(angle)f(will)f(b)q(e)j (indicated)970 57 y(b)o(y)e(the)g(new)g(FITS)g(k)o(eyw)o(ord)f Fp(LONGPOLE)f Fw(and)i(is)f(understo)q(o)q(d)970 106 y(to)e(ha)o(v)o(e)f(a)g(default)h(v)n(alue)f(of)g(0)1477 91 y Fj(\016)1511 106 y Fw(or)h(180)1627 91 y Fj(\016)1661 106 y Fw(where)g(p)q(ossible)g(\(see)970 156 y(b)q(elo)o(w\).)d(The)i (form)o(ul\032)c(for)j(the)g(co)q(ordinate)h(rotation)e(are)970 245 y(sin)7 b Fo(\022)72 b Fw(=)21 b(sin)7 b Fo(\016)i Fw(sin)e Fo(\016)1333 251 y Fi(P)1370 245 y Fw(+)i(cos)f Fo(\016)h Fw(cos)e Fo(\016)1581 251 y Fi(P)1616 245 y Fw(cos)q(\()p Fo(\013)i Fn(\000)h Fo(\013)1793 251 y Fi(P)1820 245 y Fw(\))970 295 y Fo(\036)f Fn(\000)h Fo(\036)1071 301 y Fi(P)1119 295 y Fw(=)21 b(arg)7 b([sin)g Fo(\016)h Fw(cos)g Fo(\016)1414 301 y Fi(P)1451 295 y Fn(\000)i Fw(cos)d Fo(\016)i Fw(sin)e Fo(\016)1658 301 y Fi(P)1693 295 y Fw(cos)q(\()p Fo(\013)i Fn(\000)g Fo(\013)1869 301 y Fi(P)1897 295 y Fw(\))p Fo(;)1181 345 y Fn(\000)h Fw(cos)d Fo(\016)i Fw(sin\()p Fo(\013)g Fn(\000)h Fo(\013)1484 351 y Fi(P)1511 345 y Fw(\)])d Fo(;)1957 295 y Fw(\(4\))970 432 y(where)16 b(arg)e(is)h(an)f(in)o(v)o(erse)h(tangen)o(t)g(function) f(that)h(returns)h(the)970 482 y(correct)22 b(quadran)o(t,)e Fr(i.e.)p Fw(,)f(if)h(\()p Fo(x;)7 b(y)q Fw(\))23 b(=)g(\()p Fo(r)8 b Fw(cos)f Fo(\013;)g(r)g Fw(sin)g Fo(\013)p Fw(\),)20 b(then)970 532 y(arg)7 b(\()p Fo(x;)g(y)q Fw(\))18 b(=)g Fo(\013)p Fw(.)f(The)h(longitude)e(and)i(latitude)f(in)g(the)h(nativ)o (e)970 582 y(system)13 b(are)g(giv)o(en)f(b)o(y)h(\()p Fo(\036;)7 b(\022)q Fw(\))13 b(and)f(in)h(the)g(standard)g(system)g (are)970 632 y(giv)o(en)c(b)o(y)h(\()p Fo(\013;)d(\016)r Fw(\).)h(The)i(co)q(ordinates)g(\()p Fo(\013)1586 638 y Fi(P)1614 632 y Fo(;)d(\016)1651 638 y Fi(P)1678 632 y Fw(\))j(are)g(the)g(standard-)970 681 y(system)20 b(co)q(ordinates)g (of)f(the)h(North)g(P)o(ole)g(of)f(the)h(\()p Fo(\036;)7 b(\022)q Fw(\))20 b(sys-)970 731 y(tem)c(and)g Fo(\036)1164 737 y Fi(P)1208 731 y Fw(is)g(the)h(longitude)e(in)h(the)h(nativ)o(e)f (system)g(of)g(the)970 781 y(North)10 b(P)o(ole)g(of)g(the)g(standard)h (system)e(\(sp)q(eci\014ed)j(with)e(k)o(eyw)o(ord)970 831 y Fp(LONGPOLE)o Fw(\).)j(Note)h(that,)g(if)f Fo(\016)1444 837 y Fi(p)1475 831 y Fw(=)f(90)1561 816 y Fj(\016)1579 831 y Fw(,)i(Eq.)e(4)i(b)q(ecomes)974 920 y Fo(\022)23 b Fw(=)e Fo(\016)970 982 y(\036)g Fw(=)g Fo(\036)1094 988 y Fi(p)1122 982 y Fw(+)9 b Fo(\013)g Fn(\000)h Fo(\013)1268 988 y Fi(p)1296 982 y Fw(+)f(180)1400 965 y Fj(\016)970 1072 y Fw(whic)o(h)j(ma)o(y)f(b)q(e)i(used)g(to)g(de\014ne)g(a)f (simple)f(c)o(hange)i(in)f(the)h(origin)970 1122 y(of)g(longitude.)g(A) h(useful)g(v)n(arian)o(t)f(of)g(Eqs.)g(4)g(is)970 1208 y(sin)7 b Fo(\022)13 b Fw(=)f(sin)7 b Fo(\016)i Fw(sin)e Fo(\016)1265 1214 y Fi(P)1302 1208 y Fw(+)j(cos)d Fo(\016)i Fw(cos)f Fo(\016)1514 1214 y Fi(P)1549 1208 y Fw(cos\()p Fo(\013)h Fn(\000)h Fo(\013)1725 1214 y Fi(P)1752 1208 y Fw(\))970 1258 y(cos)e Fo(\022)g Fw(sin\()p Fo(\036)h Fn(\000)h Fo(\036)1228 1264 y Fi(P)1255 1258 y Fw(\))i(=)g Fn(\000)7 b Fw(cos)g Fo(\016)i Fw(sin\()p Fo(\013)g Fn(\000)h Fo(\013)1627 1264 y Fi(P)1654 1258 y Fw(\))970 1308 y(cos)e Fo(\022)g Fw(cos)q(\()p Fo(\036)h Fn(\000)g Fo(\036)1232 1314 y Fi(P)1260 1308 y Fw(\))j(=)f(sin)c Fo(\016)i Fw(cos)f Fo(\016)1497 1314 y Fi(P)1329 1358 y Fn(\000)i Fw(cos)d Fo(\016)i Fw(sin)e Fo(\016)1536 1364 y Fi(P)1571 1358 y Fw(cos\()p Fo(\013)j Fn(\000)f Fo(\013)1747 1364 y Fi(P)1774 1358 y Fw(\))e Fo(:)1957 1283 y Fw(\(5\))970 1445 y(The)15 b(in)o(v)o(erse)f(equations)g(are)970 1532 y(sin)7 b Fo(\016)14 b Fw(=)d(sin)c Fo(\022)i Fw(sin)e Fo(\016)1265 1538 y Fi(P)1302 1532 y Fw(+)j(cos)d Fo(\022)i Fw(cos)e Fo(\016)1514 1538 y Fi(P)1549 1532 y Fw(cos)q(\()p Fo(\036)i Fn(\000)g Fo(\036)1721 1538 y Fi(P)1749 1532 y Fw(\))970 1582 y(cos)f Fo(\016)h Fw(sin\()p Fo(\013)g Fn(\000)g Fo(\013)1231 1588 y Fi(P)1258 1582 y Fw(\))j(=)g Fn(\000)7 b Fw(cos)h Fo(\022)g Fw(sin\()p Fo(\036)h Fn(\000)h Fo(\036)1627 1588 y Fi(P)1654 1582 y Fw(\))970 1632 y(cos)e Fo(\016)h Fw(cos\()p Fo(\013)g Fn(\000)h Fo(\013)1236 1638 y Fi(P)1263 1632 y Fw(\))i(=)g(sin)6 b Fo(\022)j Fw(cos)e Fo(\016)1500 1638 y Fi(P)1332 1681 y Fn(\000)j Fw(cos)d Fo(\022)i Fw(sin)e Fo(\016)1540 1687 y Fi(P)1575 1681 y Fw(cos\()p Fo(\036)i Fn(\000)h Fo(\036)1747 1687 y Fi(P)1774 1681 y Fw(\))d Fo(:)1957 1607 y Fw(\(6\))970 1769 y(A)13 b(matrix)e(metho)q(d)h(of)g(handling)g(this)h(rotation)f (is)g(describ)q(ed)j(in)970 1819 y(App)q(endix)e(D.)1032 1868 y(In)k(some)e(of)h(the)h(pro)r(jections,)f(the)h(reference)i(p)q (oin)o(t)c(is)i(not)970 1918 y(at)11 b(the)i(North)e(p)q(ole)h(of)e (the)i(nativ)o(e)g(co)q(ordinates,)f(but)h(instead)g(at)970 1968 y(some)f(other)i(latitude,)e(namely)f(\()p Fo(\036;)d(\022)q Fw(\))k(=)h(\(0)p Fo(;)7 b(\022)1711 1974 y Fu(0)1730 1968 y Fw(\),)k(whic)o(h)h(w)o(e)g(will)970 2018 y(represen)o(t)17 b(b)o(y)d(\()p Fo(\013)1251 2024 y Fu(0)1269 2018 y Fo(;)7 b(\016)1306 2024 y Fu(0)1324 2018 y Fw(\))15 b(in)e(the)i(standard)g (system.)e(The)i(default)970 2068 y(v)n(alue)c(of)g Fp(LONGPOLE)f Fw(is)i(0)1367 2053 y Fj(\016)1397 2068 y Fw(if)f(the)h(latitude)g(in)f (the)h(standard)g(sys-)970 2118 y(tem)f(of)f(the)i(reference)i(p)q(oin) o(t)c(of)h(the)h(pro)r(jection)f(is)g(greater)i(than)970 2167 y(the)i(latitude)f(of)f(that)i(p)q(oin)o(t)f(in)f(the)i(nativ)o(e) f(system;)g(otherwise)970 2217 y(the)d(default)e(v)n(alue)h(of)f Fp(LONGPOLE)f Fw(is)i(180)1605 2202 y Fj(\016)1623 2217 y Fw(.)g(This)g(is)g(the)g(condition)970 2267 y(for)k(the)h(celestial)f (latitude)g(to)f(increase)j(in)d(the)i(same)e(direction)970 2317 y(as)g(the)g(nativ)o(e)f(latitude)g(at)h(the)g(reference)i(p)q (oin)o(t.)d(Th)o(us,)g(for)g(ex-)970 2367 y(ample,)i(in)i(zenithal)h (pro)r(jections)g(the)g(default)f(is)g(alw)o(a)o(ys)f(180)1991 2352 y Fj(\016)970 2416 y Fw(since)k Fo(\022)1095 2422 y Fu(0)1133 2416 y Fw(=)f(90)1225 2401 y Fj(\016)1244 2416 y Fw(.)g(In)g(cylindrical)f(and)h(standard)g(pro)r(jections,)970 2466 y(where)e Fo(\022)1110 2472 y Fu(0)1141 2466 y Fw(=)c(0)1206 2451 y Fj(\016)1225 2466 y Fw(,)i(the)g(default)g(v)n(alue)g(for)g Fp(LONGPOLE)e Fw(is)i(180)1927 2451 y Fj(\016)1960 2466 y Fw(for)970 2516 y Fo(\016)988 2522 y Fu(0)1019 2516 y Fo(<)e Fw(0,)h(but)h(it)f(is)h(0)1289 2501 y Fj(\016)1322 2516 y Fw(for)f Fo(\016)1403 2522 y Fu(0)1434 2516 y Fo(>)f Fw(0.)1032 2566 y(The)j(co)q(ordinates)f(of)g(the)g(p)q(ole)g (ma)o(y)e(b)q(e)i(computed)g(using,)1162 2655 y Fo(\016)1180 2661 y Fi(P)1229 2655 y Fw(=)21 b(arg)7 b(\(cos)h Fo(\022)1445 2661 y Fu(0)1471 2655 y Fw(cos)f Fo(\036)1558 2661 y Fi(p)1577 2655 y Fo(;)14 b Fw(sin)7 b Fo(\022)1680 2661 y Fu(0)1699 2655 y Fw(\))21 b Fn(\006)p eop %%Page: 5 5 5 4 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)448 b(5)195 -62 y Fw(cos)251 -79 y Fj(\000)p Fu(1)302 -133 y Fl( )526 -90 y Fw(sin)7 b Fo(\016)602 -84 y Fu(0)p 340 -71 467 2 v 340 -63 a Fl(p)p 382 -63 426 2 v 39 x Fw(sin)433 -41 y Fu(2)458 -24 y Fo(\022)477 -18 y Fu(0)506 -24 y Fw(+)i(cos)603 -36 y Fu(2)628 -24 y Fo(\022)647 -18 y Fu(0)673 -24 y Fw(cos)729 -36 y Fu(2)754 -24 y Fo(\036)779 -18 y Fi(P)812 -133 y Fl(!)870 -62 y Fw(\(7\))-112 38 y(sin\()p Fo(\013)-18 44 y Fu(0)10 38 y Fn(\000)g Fo(\013)78 44 y Fi(P)105 38 y Fw(\))21 b(=)g(sin)7 b Fo(\036)278 44 y Fi(P)313 38 y Fw(cos)g Fo(\022)394 44 y Fu(0)413 38 y Fo(=)g Fw(cos)g Fo(\016)521 44 y Fu(0)870 38 y Fw(\(8\))-116 122 y(cos\()p Fo(\013)-18 128 y Fu(0)10 122 y Fn(\000)i Fo(\013)78 128 y Fi(P)105 122 y Fw(\))21 b(=)200 94 y(sin)7 b Fo(\022)277 100 y Fu(0)306 94 y Fn(\000)i Fw(sin)e Fo(\016)423 100 y Fi(P)458 94 y Fw(sin)g Fo(\016)534 100 y Fu(0)p 200 112 353 2 v 269 150 a Fw(cos)h Fo(\016)350 156 y Fi(P)385 150 y Fw(cos)f Fo(\016)465 156 y Fu(0)565 122 y Fo(;)293 b Fw(\(9\))-116 231 y(after)13 b(whic)o(h)h(Eqs.)e(4)i(or)f(5)h(ma)o(y)d(b)q(e)k(used)f(to)g (determine)f(the)h(na-)-116 281 y(tiv)o(e)j(co)q(ordinates.)h(Note)g (that)g(Eq.)12 b(7)17 b(con)o(tains)h(an)f(am)o(biguit)o(y)-116 331 y(in)11 b(the)h(sign)g(of)f(the)i(in)o(v)o(erse)f(cosine)h(and)e (that)h(all)f(three)i(indicate)-116 380 y(that)i(some)f(com)o (binations)f(of)h Fo(\013)409 386 y Fu(0)427 380 y Fw(,)h Fo(\016)472 386 y Fu(0)491 380 y Fw(,)f Fo(\022)536 386 y Fu(0)556 380 y Fw(,)g(and)h Fo(\036)689 386 y Fi(P)731 380 y Fw(are)h(not)f(al-)-116 430 y(lo)o(w)o(ed.)e(F)m(or)g(these)j (pro)r(jections,)e(w)o(e)h(are)f(therefore)i(required)f(to)-116 480 y(adopt)e(the)i(additional)d(con)o(v)o(en)o(tions:)-103 546 y(1.)27 b(Equations.)12 b(8)f(and)g(9)g(indicate)g(that)g Fo(\013)573 552 y Fi(p)603 546 y Fw(is)g(unde\014ned)h(when)-54 596 y Fo(\016)-36 602 y Fu(0)-1 596 y Fw(=)k Fn(\006)p Fw(90)121 581 y Fj(\016)140 596 y Fw(.)g(This)g(simply)f(represen)o(ts) k(the)e(longitude)f(sin-)-54 646 y(gularit)o(y)c(at)h(the)h(p)q(ole)g (and)f(forces)h(us)g(to)f(de\014ne)i Fo(\013)756 652 y Fi(p)786 646 y Fw(=)d Fo(\013)857 652 y Fu(0)888 646 y Fw(in)-54 696 y(this)i(case.)-103 746 y(2.)28 b(If)13 b(the)g(North)g(p)q(ole)g(of)f(the)h(nativ)o(e)f(system)h(approac)o (hes)g(the)-54 796 y(North)k(or)g(South)g(p)q(ole)g(of)f(the)i (standard)f(system,)g(namely)-54 845 y Fo(\016)-36 851 y Fi(p)-3 845 y Fw(=)c Fn(\006)p Fw(90)116 830 y Fj(\016)135 845 y Fw(,)i(then)g Fo(\016)275 851 y Fu(0)308 845 y Fw(=)f Fo(\022)373 851 y Fu(0)392 845 y Fw(.)g(The)i(longitude)e(at)h (the)g(nativ)o(e)-54 895 y(p)q(ole)j(is)g(then)h Fo(\013)211 901 y Fi(p)249 895 y Fw(=)g Fo(\013)327 901 y Fu(0)357 895 y Fw(+)13 b Fo(\036)427 901 y Fi(p)458 895 y Fn(\000)f Fw(180)565 880 y Fj(\016)602 895 y Fw(for)18 b Fo(\016)688 901 y Fi(p)726 895 y Fw(=)h(90)819 880 y Fj(\016)856 895 y Fw(and)-54 945 y Fo(\013)-27 951 y Fi(p)3 945 y Fw(=)12 b Fo(\013)74 951 y Fu(0)101 945 y Fn(\000)e Fo(\036)168 951 y Fi(p)201 945 y Fw(for)j Fo(\016)282 951 y Fi(p)313 945 y Fw(=)f Fn(\000)p Fw(90)431 930 y Fj(\016)450 945 y Fw(.)-103 995 y(3.)32 b(Some)15 b(com)o(binations)f(of)i Fo(\022)399 1001 y Fu(0)418 995 y Fw(,)g Fo(\016)464 1001 y Fu(0)483 995 y Fw(,)g(and)g Fo(\036)619 1001 y Fi(p)654 995 y Fw(pro)q(duce)h(an)g(in-)-54 1045 y(v)n(alid)f(argumen)o (t)g(for)h(the)h Fo(cos)438 1030 y Fj(\000)p Fu(1)483 1045 y Fw(\(\))g(in)f(Eq.)12 b(7.)17 b(In)g(this)h(case)-54 1094 y(there)g(is)e(no)h(solution)f(for)g Fo(\016)404 1100 y Fi(p)424 1094 y Fw(.)g(Otherwise)i(Eq.)12 b(7)k(pro)q(duces)-54 1144 y(t)o(w)o(o)g(solutions)g(for)g Fo(\016)287 1150 y Fi(p)307 1144 y Fw(.)g(V)m(alid)f(solutions)h(are)h(ones)g(that)f (lie)-54 1194 y(in)e(the)h(range)f Fn(\000)p Fw(90)254 1179 y Fj(\016)287 1194 y Fw(to)g(+90)412 1179 y Fj(\016)431 1194 y Fw(.)g(F)m(or)g(some)f(com)o(binations)f(of)-54 1244 y Fo(\022)-35 1250 y Fu(0)-16 1244 y Fw(,)f Fo(\016)25 1250 y Fu(0)44 1244 y Fw(,)h(and)f Fo(\036)171 1250 y Fi(p)190 1244 y Fw(,)g(neither)i(solution)e(is)h(v)n(alid.)d(Note,)j (ho)o(w)o(ev)o(er,)-54 1294 y(that)j(when)h Fp(LONGPOLE)d Fw(\()p Fn(\021)h Fo(\036)423 1300 y Fi(p)442 1294 y Fw(\))h(tak)o(es)h(its)f(default)g(v)n(alue)f(of)-54 1343 y(0)-33 1328 y Fj(\016)-1 1343 y Fw(or)f(180)112 1328 y Fj(\016)143 1343 y Fw(\(dep)q(ending)h(on)f Fo(\022)434 1349 y Fu(0)453 1343 y Fw(\))g(then)h Fr(any)g Fw(com)o(bination)c(of) -54 1393 y Fo(\016)-36 1399 y Fu(0)-4 1393 y Fw(and)k Fo(\022)96 1399 y Fu(0)128 1393 y Fw(pro)q(duces)h(a)e(v)n(alid)f (argumen)o(t)g(to)i(the)g(cos)798 1378 y Fj(\000)p Fu(1)843 1393 y Fw(\(\))f(in)-54 1443 y(Eq.)f(7,)h(and)h(at)g(least)g(one)g(of)f (the)i(solutions)e(is)h(v)n(alid.)-103 1493 y(4.)28 b(Where)12 b(Eq.)g(7)g(has)g(t)o(w)o(o)f(v)n(alid)f(solutions)i(the)g(one)g (closest)h(to)-54 1543 y(the)f(v)n(alue)g(of)f(the)i(new)f(FITS)g(k)o (eyw)o(ord)g Fp(LATPOLE)f Fw(is)g(c)o(hosen.)-54 1593 y(It)18 b(is)h(acceptable)g(to)g(set)g Fp(LATPOLE)e Fw(to)i(a)f(n)o(um) o(b)q(er)g(greater)-54 1642 y(than)12 b(+90)g(to)g(c)o(ho)q(ose)h(the)g (northerly)g(solution)e(\(the)i(default)-54 1692 y(if)h Fp(LATPOLE)g Fw(is)h(omitted\),)f(or)h(a)g(n)o(um)o(b)q(er)g(less)h (than)f Fn(\000)p Fw(90)g(to)-54 1742 y(select)g(the)g(southern)g (solution.)-103 1792 y(5.)32 b(Eq.)12 b(7)17 b(often)f(only)g(has)h (one)f(v)n(alid)f(solution)h(\(b)q(ecause)i(the)-54 1842 y(other)11 b(lies)g(outside)g(the)h(range)f Fn(\000)p Fw(90)513 1827 y Fj(\016)543 1842 y Fw(to)g(+90)665 1827 y Fj(\016)683 1842 y Fw(\).)g(In)g(this)g(case)-54 1891 y Fp(LATPOLE)h Fw(is)i(ignored.)-103 1941 y(6.)31 b(F)m(or)15 b(the)h(sp)q(ecial)g(case)h(where)f Fo(\022)477 1947 y Fu(0)511 1941 y Fw(=)e(0,)h Fo(\016)623 1947 y Fu(0)656 1941 y Fw(=)g(0,)f(and)i Fo(\036)858 1947 y Fi(p)891 1941 y Fw(=)-54 1991 y Fn(\006)p Fw(90)11 b(then)h Fo(\016)141 1997 y Fi(p)171 1991 y Fw(is)f(not)g(determined)h(and)f Fp(LATPOLE)e Fw(sp)q(eci\014es)k(it)-54 2041 y(completely)m(.)f Fp(LATPOLE)g Fw(has)i(no)g(default)f(v)n(alue)g(in)h(this)g(case.)-116 2107 y(In)c(ligh)o(t)f(of)g(the)i(ab)q(o)o(v)o(e)f(p)q(oin)o(ts,)f (users)j(should)e(c)o(hec)o(k)h(their)f(v)n(alues)-116 2157 y(of)15 b Fo(\022)-48 2163 y Fu(0)-29 2157 y Fw(,)g Fo(\016)16 2163 y Fu(0)35 2157 y Fw(,)g(and)g Fo(\036)169 2163 y Fi(p)204 2157 y Fw(against)g(Eqs.)d(7,)j(8,)g(and)g(9)h(to)f (ensure)i(their)-116 2207 y(v)n(alidit)o(y)m(.)-54 2257 y(This)h(prop)q(osal)g(eliminates)e(the)j(need)g(for)f(the)h (previously)-116 2307 y(sp)q(eci\014ed)k(k)o(eyw)o(ord)f Fp(CROTA)p Fr(n)p Fw(.)e(In)i(Section)13 b(5)21 b(w)o(e)h(will)f (discuss)-116 2356 y(considerations)15 b(in)f(the)h(translation)f(of)g (older)g(FITS)h(images)e(to)-116 2406 y(the)e(new)g(system)g(and)g(pro) o(visions)f(that)h(up)q(dated)h(FITS-writing)-116 2456 y(programs)d(should)i(mak)o(e)e(to)h(supp)q(ort)i(older)f(FITS-reading) f(pro-)-116 2506 y(grams.)-54 2556 y(The)24 b(k)o(eyw)o(ords)h(prop)q (osed)g(ab)q(o)o(v)o(e)f(and)g(throughout)g(the)-116 2605 y(main)11 b(b)q(o)q(dy)i(of)f(this)h(man)o(uscript)f(apply)h(to)g (the)g(relativ)o(ely)g(sim-)-116 2655 y(ple)18 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2144 y Fp(C)p Fr(m)s Fp(PIX)p Fr(n)50 b Fw(pixel)13 b(co)q(ordinate)h(of)g(the)g(reference)i(p)q(oin)o(t)1001 2194 y Fp(C)p Fr(m)s Fp(ELT)p Fr(n)50 b Fw(co)q(ordinate)14 b(incremen)o(t)g(at)f(reference)k(p)q(oin)o(t)1001 2244 y Fp(C)p Fr(m)s Fp(YPE)p Fr(n)50 b Fw(axis)13 b(t)o(yp)q(e)i(\(8)e(c)o (haracters\))1001 2294 y Fp(C)p Fr(m)s Fp(NIT)p Fr(n)50 b Fw(units)14 b(of)f Fp(C)p Fr(m)s Fp(VAL)p Fr(n)h Fw(and)f Fp(C)p Fr(m)s Fp(ELT)p Fr(n)h Fw(\(c)o(haracter\))970 2356 y(The)f(v)n(alue)e(of)h Fr(m)g Fw(will)e(b)q(e)j(2)f(through)g(9)g (for)g(the)g(second)h(through)970 2406 y(nin)o(th)j(description)g(of)g (axis)f Fr(n)p Fw(.)h(The)g(alternate)h(co)q(ordinate)f(de-)970 2456 y(scriptions)j(are)f(computed)g(in)g(the)g(same)g(fashion)f(as)h (the)h(pri-)970 2506 y(mary)e(co)q(ordinates.)i(Note)g(that)g(the)g(t)o (yp)q(e)g(of)f(co)q(ordinate)h(de-)970 2556 y(p)q(ends)14 b(on)e(the)h Fp(CTYPE)p Fr(n)e Fw(or)h Fp(C)p Fr(m)p Fp(YPE)p Fr(n)g Fw(and)g(ma)o(y)f(b)q(e)i(linear)e(in)h(one)970 2605 y(of)19 b(the)i(alternate)f(descriptions)h(and)e(non-linear)g(in)h (another.)970 2655 y(\(Con)o(v)o(en)o(tions)h(implied)d(b)o(y)i(sp)q (ecial)h(v)n(alues)g(for)f Fp(CTYPE)p Fr(n)g Fw(and)p eop %%Page: 6 6 6 5 bop -116 -192 a Fs(6)449 b(Greisen)14 b(&)e(Calabretta:)i(Represen) o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 -93 y Fp(C)p Fr(m)p Fp(YPE)p Fr(n)9 b Fw(for)h(describing)g (non-celestial)g(non-linear)f(co)q(ordinates)-116 -43 y(will)h(need)i(to)g(b)q(e)g(dev)o(elop)q(ed,)g(but)f(are)h(b)q(ey)o (ond)g(the)h(scop)q(e)f(of)f(the)-116 7 y(presen)o(t)i(man)o (uscript.\))d(Note)i(to)q(o)f(that)g(w)o(e)h(do)f(not)g(prop)q(ose)h (an)o(y)-116 57 y(defaults)e(for)g(these)i(k)o(eyw)o(ords;)f(if)e(a)i (secondary)g(axis)f(description)-116 106 y(is)j(giv)o(en,)g(all)g(of)g (these)i(k)o(eyw)o(ords)g(m)o(ust)e(b)q(e)h(sp)q(eci\014ed.)-116 222 y Fr(2.4.)g(Image)i(display)e(c)n(onventions)-116 302 y Fw(It)g(is)g(v)o(ery)h(helpful)e(to)h(adopt)g(a)g(con)o(v)o(en)o 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y(new)h(\(p)q(ost-IA)o(U)g(1976\))f(system)-52 2111 y Fp('GAPPT)20 b(')93 b Fw(geo)q(cen)o(tric)15 b(apparen)o(t)g(place,)263 2161 y(p)q(ost-IA)o(U)f(1976)f(system)-54 2257 y(The)18 b(initial)e(FITS)j(pap)q(er)f(\(W)m(ells,)f Fr(et)i(al)p Fw(,)e(1981)o(\))h(used)h(the)-116 2307 y(k)o(eyw)o(ord)e Fp(EPOCH)g Fw(to)g(mean)g(the)h(ep)q(o)q(c)o(h)h(of)e(the)h(mean)f (equator)-116 2356 y(and)12 b(equino)o(x.)f(Since)h(the)h(w)o(ord)f Fr(ep)n(o)n(ch)h Fw(has)f(a)g(con)o(v)o(en)o(tional)f(and)-116 2406 y(di\013eren)o(t)19 b(use)h(in)e(astrometry)m(,)g(w)o(e)h(prop)q (ose)g(replacing)g Fp(EPOCH)-116 2456 y Fw(with)e Fp(EQUINOX)o Fw(.)h(If)g(the)g(old)g(k)o(eyw)o(ord)g(is)g(giv)o(en,)f(it)h(should)g (b)q(e)-116 2506 y(replaced)h(with)e(the)i(new)g(one.)e(F)m(or)h Fp(FK4)g Fw(and)f Fp(FK4-NO-E)p Fw(,)f(an)o(y)-116 2556 y(stated)k(equino)o(x)e(is)h(Besselian)h(and,)f(if)f(neither)i Fp(EQUINOX)e Fw(nor)-116 2605 y Fp(EPOCH)e Fw(are)i(giv)o(en,)f(a)h 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Fw(k)o(eyw)o(ord,)970 306 y(are)i(not)f(required)h(of)f(them.)f(Note)h(that)g Fp(EQUINOX)f Fw(is)h(understo)q(o)q(d)970 356 y(to)g(apply)f(to)h (ecliptic)g(co)q(ordinates)h(in)f(the)g(standard)h(w)o(a)o(y)e(as)h(w)o (ell)970 405 y(as)k(to)g(equatorial)f(co)q(ordinates.)1032 470 y(The)20 b(FK4)f(reference)j(system)d(is)g(not)g(inertial;)e(there) k(is)e(a)970 519 y(small,)11 b(but)j(signi\014can)o(t,)e(rotation)h (relativ)o(e)g(to)g(distan)o(t)h(ob)r(jects.)970 569 y(Therefore,)h(w)o(e)e(require)i(an)e(ep)q(o)q(c)o(h)i(to)e(sp)q(ecify) h(when)g(the)h(mean)970 619 y(place)c(w)o(as)f(correct)j(as)d(w)o(ell)g (as)h(the)g(equino)o(x)f(whic)o(h)g(sp)q(eci\014es)j(the)970 669 y(reference)k(frame.)12 b(The)j(old)f(k)o(eyw)o(ord)g Fp(DATE-OBS)e Fw(ma)o(y)h(b)q(e)i(used)970 719 y(for)c(this)f(purp)q (ose.)i(T)m(o)e(pro)o(vide)h(a)f(more)g(accurate,)h(and)g(easier)h(to) 970 769 y(use,)h(sp)q(eci\014cation,)f(w)o(e)g(also)f(prop)q(ose)i(the) g(new)f(\015oating-v)n(alued)970 818 y(k)o(eyw)o(ord)21 b Fp(MJD-OBS)e Fw(to)h(sp)q(ecify)h(the)h(time)d(of)h(of)g(observ)n (ation)970 868 y(in)15 b(the)h(form)e(of)h(a)g(Mo)q(di\014ed)h(Julian)f (Date)g(\(JD)c Fn(\000)f Fw(2400000)p Fo(:)p Fw(5\).)970 918 y(Of)21 b(necessit)o(y)m(,)g(this)g(time)f(m)o(ust)f(represen)o(t)k (some)d(a)o(v)o(erage)h(of)970 968 y(the)g(times)d(actually)h(en)o (tering)h(the)h(observ)n(ation,)e(but)h(should)970 1018 y(b)q(e)c(used)h(in)e(preference)j(to)e Fp(DATE-OBS)d Fw(when)j(a)o(v)n(ailable.)d(Com-)970 1067 y(men)o(ts)g(explaining)f (the)i(nature)g(of)e(this)i(\\a)o(v)o(erage")e(time)g(w)o(ould)970 1117 y(b)q(e)17 b(quite)g(appropriate.)f(F)m(uture)h(agreemen)o(ts)f (ma)o(y)e(clarify)i(the)970 1167 y(time)c(system)g(used)i(to)e (determine)h(mo)q(di\014ed)e(Julian)h(dates,)h(but)970 1217 y(the)h(usage)f(of)g Fp(MJD-OBS)e Fw(should)i(not)g(require)h(suc) o(h)g(detailed)f(ac-)970 1267 y(curacy)g(due)f(to)g(the)g(\\a)o(v)o (erage")g(nature)g(of)f(the)i(time)d(giv)o(en.)h(The)970 1316 y Fp(GAPPT)g Fw(system)g(of)g(co)q(ordinates)i(do)q(es)f(not)g (require)g(an)f Fp(EQUINOX)p Fw(,)970 1366 y(but)j(only)f(an)h Fp(MJD-OBS)p Fw(.)1032 1431 y(The)30 b(com)o(bination)d(of)h(the)i Fp(EQUINOX)p Fw(,)e Fp(RADECSYS)p Fw(,)f(and)970 1480 y Fp(CTYPE)p Fr(n)10 b Fw(FITS)h(header)g(cards)h(de\014ne)f(the)h(co)q (ordinate)f(system)f(in)970 1530 y(whic)o(h)j(the)h Fp(CRVAL)p Fr(n)e Fw(are)h(sp)q(eci\014ed.)h(It)f(is)g(also)f(understo)q(o)q(d)i (that)970 1580 y(the)f(FITS)f(image)f(has)h(b)q(een)h(resampled)f(so)h (as)f(to)g(con)o(triv)o(e)g(that)970 1630 y(application)18 b(of)h(the)g(algorithm)e(for)i(computing)e(image)g(co)q(or-)970 1680 y(dinates)g(\(Fig.)11 b(1\))16 b(will)f(pro)q(duce)j(co)q (ordinates)e(of)g(the)h(sp)q(eci\014ed)970 1730 y(t)o(yp)q(e)h(at)g (all)e(p)q(oin)o(ts)h(of)g(the)h(image.)e(This)h(resampling)f(will)g(b) q(e)970 1779 y(imp)q(ortan)o(t)11 b(for)i(applications)e(requiring)i (astrometric)f(accuracy)970 1829 y(\(for)k(example)f(in)o 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b(celestial)h(co)q(ordinates,)g(w)o(e)g(deem)f(that)h(the)h(units)f (for)f(all)-116 788 y(measures)f(of)g(angles)g(on)g(the)h(celestial)g (sphere)h(purp)q(orting)e(to)-116 837 y(b)q(e)d(within)e(the)i(con)o(v) o(en)o(tions)g(of)e(the)i(presen)o(t)h(man)o(uscript)e(shall)-116 887 y(b)q(e)h(\(are)h(required)f(to)g(b)q(e,)g(m)o(ust)f(b)q(e\))i (degrees.)-54 943 y(The)c(units)g(of)f(the)i Fp(CDELT)p Fr(n)e Fw(are)h(degrees)i(p)q(er)f(pixel)e(for)g(angu-)-116 993 y(lar)k(axes,)h(making)d(degrees)17 b(the)e(units)g(of)f(\()p Fo(x;)7 b(y)q Fw(\),)15 b(and)g Fo(R)807 999 y Fi(\022)825 993 y Fw(.)f(The)-116 1043 y(units)c(of)g(\()p Fo(\036;)d(\022)q Fw(\))k(and)f Fo(A)243 1049 y Fi(\036)276 1043 y Fw(are)h(also)e (degrees)k(simply)8 b(b)q(ecause)k(they)-116 1093 y(are)i(angles.)f(F)m (rom)f(the)i(rev)o(erse)i(p)q(oin)o(t)d(of)g(view,)g(\()p Fo(x;)7 b(y)q Fw(\),)14 b(and)g Fo(R)905 1099 y Fi(\022)-116 1142 y Fw(are)h(in)g(units)g(of)f(fractions)h(of)g(the)h(unit)f (sphere,)h(m)o(ultiplied)c(b)o(y)-111 1176 y Fu(180)-60 1163 y Fh(\016)p -111 1183 67 2 v -88 1207 a Fi(\031)-24 1192 y Fw(to)j(bring)g(the)h(units)f(to)g(the)h(FITS-con)o(v)o(en)o (tional)e(degrees)-116 1242 y(rather)k(than)f(radians.)f(In)h(the)h (form)o(ul\032)d(dev)o(elop)q(ed)j(b)q(elo)o(w,)e(it)-116 1292 y(is)f(assumed)g(that)h(all)e(trigonometric)g(functions)i(require) g(argu-)-116 1342 y(men)o(ts)g(in)h(degrees)i(and)f(all)e(in)o(v)o (erse)i(trigonometric)e(functions)-116 1392 y(return)11 b(v)n(alues)e(in)h(degrees.)h(The)f(argumen)o(ts)f(of)g(in)o(v)o(erse)i (trigono-)-116 1441 y(metric)e(functions)i(and)f(the)h(results)h (returned)g(b)o(y)e(trigonometric)-116 1491 y(functions)i(are)h (assumed)f(to)g(tak)o(e)g(their)h(normal)d(v)n(alues.)i(When-)-116 1541 y(ev)o(er)h(a)g(con)o(v)o(ersion)g(b)q(et)o(w)o(een)h(radians)f (and)g(degrees)h(is)f(required)-116 1591 y(in)g(the)h(equations)g(b)q (elo)o(w,)g(it)f(is)h(explicitly)f(sho)o(wn.)-54 1647 y(The)g(equations)g(b)q(elo)o(w)f(describ)q(e)i(a)f Fr(left-hande)n(d)g Fw(co)q(ordinate)-116 1697 y(system,)22 b(called)g(\()p Fo(x;)7 b(y)q Fw(\),)23 b(whic)o(h)f(will)f(b)q(e)j(cen)o(tered)h (either)e(on)-116 1746 y(the)d(north)g(p)q(ole)g(of)f(the)h(nativ)o(e)g (co)q(ordinate)g(system)f(\(\(longi-)-116 1796 y(tude,latitude\))c Fn(\021)f Fw(\()p Fo(\036;)7 b(\022)q Fw(\))14 b(=)g(\(0)p Fo(;)7 b Fw(90)448 1781 y Fj(\016)466 1796 y Fw(\)\))15 b(or)h(on)f(the)h(origin)e(of)g(the)-116 1846 y(nativ)o(e)19 b(system)g(\(\()p Fo(\036;)7 b(\022)q Fw(\))21 b(=)h(\(0)p Fo(;)7 b Fw(0\)\).)18 b(These)j(are)f(illustrated)f(in)-116 1896 y(Fig.)11 b(2.)-54 1952 y(In)g(most)g(of)g(the)h(pro)r(jections)g (b)q(elo)o(w,)f(w)o(e)h(ha)o(v)o(e)f(not)h(included)-116 2002 y(the)i(in\014nite)g(v)n(ariet)o(y)f(of)h(oblique)f(v)n(ariations) g(on)g(the)i(basic)f(pro-)-116 2051 y(jections)19 b(suc)o(h)h(as)g (equatorial)e(cases)i(of)f(the)h(zenithal)f(pro)r(jec-)-116 2101 y(tions,)c(whic)o(h)g(are)h(included)g(in)f(man)o(y)f(b)q(o)q(oks) i(on)f(map)f(pro)r(jec-)-116 2151 y(tions.)g(W)m(e)h(b)q(eliev)o(e)g (that)h(this)f(is)g(a)g(desirable)g(omission,)e(whic)o(h)-116 2201 y(will)c(impro)o(v)o(e)g(the)j(simplicit)o(y)d(and)i(clarit)o(y)f (of)g(the)i(data)f(comm)o(u-)-116 2251 y(nication,)j(while)h(the)h (later)f(rotation)g(from)f(the)i(nativ)o(e)f(co)q(ordi-)-116 2300 y(nates)e(of)f(the)h(pro)r(jection)g(to)f(standard,)g(recognized)i (systems)e(of)-116 2350 y(co)q(ordinates)h(pro)o(vides)g(all)f(needed)j (obliquit)o(y)m(.)10 b(There)k(are)g(inn)o(u-)-116 2400 y(merable)h(other)j(pro)r(jections)f(used)h(in)e(cartograph)o(y)h(whic) o(h)g(w)o(e)-116 2450 y(ha)o(v)o(e)g(also)g(omitted.)e(Ev)o(enden)k (\(1991)o(\))f(o\013ers)g(lo)o(v)o(ely)e(displa)o(ys)-116 2500 y(\(with)9 b(little)g(mathematics\))f(for)h(73)g(di\013eren)o(t)i (pro)r(jections,)f(some)-116 2549 y(of)j(whic)o(h)h(w)o(e)g(prop)q(ose) g(either)h(directly)f(or)g(obliquely)m(.)-54 2605 y(Calabretta)e (\(1995)o(\))g(has)g(written,)g(and)g(made)f(a)o(v)n(ailable)f(un-)-116 2655 y(der)j(a)f(GNU)h(license,)g(a)f(pac)o(k)n(age)g(of)g(C)h(and)f (of)g(F)m(ortran)h(subrou-)970 -93 y(tines)h(implem)o(en)o(ting)c(all)i (pro)r(jections)h(and)g(co)q(ordinate)g(con)o(v)o(er-)970 -43 y(sions)h(prop)q(osed)h(in)e(this)h(man)o(uscript.)970 65 y Fr(4.1.)h(Zenithal)g(pr)n(oje)n(ctions)970 141 y(Zenithal)d Fw(or)f Fr(azimuthal)g Fw(pro)r(jections)h(are)f(a)g(class)g(of)g(pro)r (jections)970 191 y(in)16 b(whic)o(h)g(the)g(surface)h(of)f(pro)r (jection)g(is)g(a)g(plane.)f(The)i(nativ)o(e)970 241 y(co)q(ordinate)j(system)g(is)f(de\014ned)i(to)f(ha)o(v)o(e)f(the)h(p)q (olar)f(axis)h(or-)970 291 y(thogonal)12 b(to)g(the)h(plane)g(of)f(pro) r(jection)h(at)f(the)i Fr(r)n(efer)n(enc)n(e)d Fw(p)q(oin)o(t.)970 341 y(Then,)19 b(the)g(meridians)e(\(of)h(nativ)o(e)h(longitude\))f (are)h(pro)r(jected)970 390 y(as)d(ra)o(ys)g(emanating)e(from)h(the)h (reference)j(p)q(oin)o(t)c(and)h(the)h(par-)970 440 y(allels)12 b(\(of)g(nativ)o(e)g(latitude\))h(are)g(mapp)q(ed)f(as)g(concen)o(tric) j(circles)970 490 y(cen)o(tered)g(on)e(the)h(same)e(p)q(oin)o(t.)g(By)i (con)o(v)o(en)o(tion,)e(therefore,)i(the)970 540 y(reference)h(p)q(oin) o(t)e(in)f(all)f(zenithal)i(pro)r(jections)g(is)g(the)g(nativ)o(e)g (co-)970 590 y(ordinate)h(system)g(north)g(p)q(ole,)f(\()p Fo(\036;)7 b(\022)q Fw(\))12 b(=)g(\(0)p Fo(;)7 b Fw(90)1735 575 y Fj(\016)1753 590 y Fw(\).)1032 640 y(De\014ning)20 b(\()p Fo(\036;)7 b(\022)q Fw(\))19 b(as)h(the)g(nativ)o(e)f(longitude) g(and)g(latitude,)970 689 y(resp)q(ectiv)o(ely)m(,)10 b(zenithal)g(pro)r(jections)g(are)g(then)g(c)o(haracterized)i(b)o(y)970 739 y(a)i(cylindrical)f(co)q(ordinate)h(radius)g Fo(R)1568 745 y Fi(\022)1600 739 y Fw(and)g(an)f(angle)h Fo(A)1878 745 y Fi(\036)1914 739 y Fw(equal)970 789 y(to)19 b(the)g(longitude)g Fo(\036)p Fw(.)f(Our)h(linear,)f(rectangular)h(co)q(ordinates,)970 839 y(\()p Fo(x;)7 b(y)q Fw(\),)14 b(de\014ned)h(b)o(y)e(Eq.)g(3,)g (are)h(then)h(giv)o(en)e(b)o(y)997 929 y Fo(x)20 b Fw(=)h Fo(R)1126 935 y Fi(\022)1151 929 y Fw(sin)7 b Fo(\036)702 b Fw(\(10\))999 992 y Fo(y)22 b Fw(=)f Fn(\000)p Fo(R)1158 998 y Fi(\022)1184 992 y Fw(cos)7 b Fo(\036)g(;)646 b Fw(\(11\))970 1054 y(whic)o(h)14 b(ma)o(y)e(b)q(e)i(in)o(v)o(erted)h (as)970 1124 y Fo(R)1002 1130 y Fi(\022)1041 1124 y Fw(=)1094 1086 y Fl(p)p 1136 1086 134 2 v 38 x Fo(x)1160 1112 y Fu(2)1187 1124 y Fw(+)10 b Fo(y)1250 1112 y Fu(2)1936 1124 y Fw(\(12\))996 1187 y Fo(\036)20 b Fw(=)h(arg)8 b(\()p Fn(\000)p Fo(y)q(;)f(x)p Fw(\))g Fo(:)629 b Fw(\(13\))970 1294 y(4.1.1.)12 b(AZP)i(\(p)q(ersp)q(ectiv)o(e\))970 1371 y(P)o(ersp)q(ectiv)o(e)28 b(zenithal)e(pro)r(jections)g(are)h (generated)g(from)d(a)970 1421 y(p)q(oin)o(t)19 b(and)g(carried)h (through)f(the)g(unit)g(sphere)i(to)e(the)h(plane)970 1471 y(of)g(pro)r(jection.)g(W)m(e)f(consider)i(here)h(only)d(those)i (cases)g(where)970 1521 y(the)13 b(plane)g(of)e(pro)r(jection)i(is)g (tangen)o(t)f(to)h(the)g(sphere)h(at)e(its)h(p)q(ole)970 1570 y(since)h(it)f(ma)o(y)e(b)q(e)j(sho)o(wn)f(that)g(ha)o(ving)f(the) i(plane)f(in)o(tersect)i(the)970 1620 y(sphere)k(at)e(another)h (\(nativ)o(e\))f(latitude)g(causes)i(only)d(a)h(simple)970 1670 y(rescaling)d(of)f(the)h(pro)r(jection.)g(If)f(the)i(source)g(of)e (the)h(pro)r(jection)970 1720 y(is)g(at)g(a)g(distance)h Fo(\026)f Fw(from)f(the)i(cen)o(ter)g(of)f(the)h(sphere)h(\()p Fo(\026)c(>)g Fw(0)i(in)970 1770 y(the)i(direction)g(a)o(w)o(a)o(y)f (from)f(the)i(plane)g(of)f(pro)r(jection\),)h(then)g(it)970 1819 y(is)e(straigh)o(tforw)o(ard)f(to)h(sho)o(w)g(that)970 1933 y Fo(R)1002 1939 y Fi(\022)1032 1933 y Fw(=)1081 1904 y(180)1144 1889 y Fj(\016)p 1081 1923 82 2 v 1109 1961 a Fo(\031)1174 1933 y Fw(cos)8 b Fo(\022)1264 1874 y Fl(\022)1329 1904 y Fo(\026)h Fw(+)g(1)p 1300 1923 155 2 v 1300 1961 a Fo(\026)g Fw(+)g(sin)e Fo(\022)1459 1874 y Fl(\023)1503 1933 y Fo(:)421 b Fw(\(14\))970 2046 y(The)15 b(in)o(v)o(erse)f(of)f(this)h(equation)g(is)971 2171 y Fo(\022)22 b Fw(=)f(arg)8 b(\()p Fo(\032;)f Fw(1\))i Fn(\000)g Fw(sin)1325 2154 y Fj(\000)p Fu(1)1377 2100 y Fl( )1468 2143 y Fo(\032\026)p 1415 2162 154 2 v 1415 2170 a Fl(p)p 1456 2170 112 2 v 1456 2206 a Fo(\032)1477 2194 y Fu(2)1506 2206 y Fw(+)g(1)1573 2100 y Fl(!)1619 2171 y Fo(;)305 b Fw(\(15\))970 2269 y(where)970 2346 y Fo(\032)21 b Fn(\021)1099 2318 y Fo(\031)p 1070 2336 82 2 v 1070 2374 a Fw(180)1133 2362 y Fj(\016)1185 2318 y Fo(R)1217 2324 y Fi(\022)p 1162 2336 97 2 v 1162 2374 a Fo(\026)9 b Fw(+)g(1)1270 2346 y Fo(:)970 2456 y Fw(This)j(pro)r (jection)g(is)g(illustrated)g(in)g(Fig.)f(3.)g(The)h(FITS)h(k)o(eyw)o (ord)970 2506 y Fp(PROJP1)f Fw(will)g(b)q(e)i(used)g(to)f(sp)q(ecify)h Fo(\026)f Fw(and)g(the)h(FITS)f(pro)r(jection)970 2556 y(t)o(yp)q(e)k(will)e(b)q(e)h(represen)o(ted)j(b)o(y)d(the)h(co)q(de)g Fp(AZP)p Fw(.)e(V)m(alues)h(of)g Fo(\026)f(>)970 2605 y Fw(10)1012 2590 y Fu(14)1057 2605 y Fw(will)9 b(b)q(e)i(tak)o(en)f (to)g(b)q(e)h(in\014nite,)f(but)g(if)f Fo(\026)j(>>)f Fw(the)g(n)o(um)o(b)q(er)e(of)970 2655 y(pixels,)k(then)h(it)g(ma)o(y)d (b)q(e)j(tak)o(en)g(to)g(b)q(e)g(in\014nite)f(with)g(v)o(ery)h(little)p eop %%Page: 8 8 8 7 bop -116 -192 a Fs(8)449 b(Greisen)14 b(&)e(Calabretta:)i(Represen) o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 733 y Fm(Fig.)g(2.)f Fs(Co)q(ordinate)h(de\014nitions)i(for)d(origins)i (at)e(\(longitude,)i(latitude\))g Fg(\021)c Fs(\()p Fk(\036;)5 b(\022)q Fs(\))11 b(=)g(\(0)p Fk(;)6 b Fs(90)1257 717 y Ff(\016)1275 733 y Fs(\))13 b(\(left\))g(and)h(\()p Fk(\036;)5 b(\022)q Fs(\))11 b(=)g(\(0)p Fk(;)6 b Fs(0\))13 b(\(righ)o(t\))-116 867 y Fw(error.)20 b(This)g(pro)r(jection)g (correctly)h(represen)o(ts)h(the)f(image)c(of)-116 916 y(a)e(sphere,)i(suc)o(h)g(as)f(a)g(planet,)f(when)i(view)o(ed)f(from)e (a)i(distance)-116 966 y Fo(\026)c Fn(\000)h Fw(1)18 b(times)f(the)j(planetary)e(radius.)g(The)h(sub-view)o(er)g(p)q(oin)o (t)-116 1016 y(is)e(then)g(the)h(south)g(p)q(ole)f(of)g(the)g(pro)r (jection,)g Fo(\022)i Fw(=)e Fn(\000)p Fw(90)808 1001 y Fj(\016)827 1016 y Fw(,)g(and)-116 1066 y(the)g(co)q(ordinates)h(of)f (the)h(reference)i(p)q(oin)o(t)c(ma)o(y)g(b)q(e)i(expressed)-116 1116 y(in)d(planetary)h(longitude)g(and)g(latitude,)f Fo(\025;)7 b(\014)r Fw(.)16 b(The)h Fp(PC)f Fw(matrix)-116 1166 y(should)g(ev)o(en)h(allo)o(w)e(for)h(\014rst-order)i(correction)g (for)e(planetary)-116 1215 y(oblateness.)-116 1408 y(4.1.2.)11 b(T)m(AN)j(\(gnomonic\))-116 1514 y(There)g(are)f(three)h(sp)q(ecial)f (v)n(alues)g(of)f Fo(\026)h Fw(in)f(p)q(ersp)q(ectiv)o(e)j(zenithal) -116 1564 y(pro)r(jections)d(whic)o(h)g(are)h(in)e(wide)h(use)h(in)f (astronom)o(y)e(and)i(whic)o(h)-116 1614 y(ha)o(v)o(e)k(come)g(to)g(ha) o(v)o(e)g(con)o(v)o(en)o(tional)g(co)q(des)h(for)f(the)h(pro)r(jection) -116 1664 y(t)o(yp)q(es.)d(If)g(y)o(ou)g(are)h(using)f(one)g(of)g(the)g (sp)q(ecial)h(v)n(alues)f(of)g Fo(\026)p Fw(,)f(y)o(ou)-116 1714 y(should)19 b(use)h(the)g(con)o(v)o(en)o(tional)f(co)q(de)h (rather)h(than)e Fp(AZP)p Fw(.)f(The)-116 1764 y(\014rst)e(of)f(these,) i Fo(\026)d Fw(=)h(0,)g(is)g(widely)g(used)i(in)e(optical)f(astronom)o (y)-116 1813 y(and)f(is)h(called)g Fp(TAN)o Fw(.)f(Equation)f(14)i(b)q (ecomes)-116 1956 y Fo(R)-84 1962 y Fi(\022)-45 1956 y Fw(=)13 1928 y(180)76 1913 y Fj(\016)p 13 1946 82 2 v 41 1984 a Fo(\031)106 1956 y Fw(cot)7 b Fo(\022)662 b Fw(\(16\))-87 2061 y Fo(\022)23 b Fw(=)e(tan)68 2044 y Fj(\000)p Fu(1)119 2002 y Fl(\022)155 2033 y Fw(180)218 2018 y Fj(\016)p 155 2051 V 158 2089 a Fo(\031)q(R)215 2095 y Fi(\022)241 2002 y Fl(\023)285 2061 y Fo(:)552 b Fw(\(17\))-116 2207 y(This)11 b(pro)r(jection)h(is)f(illustrated)h (in)f(Fig.)g(4.)g(The)h Fp(TAN)e Fw(pro)r(jection)-116 2257 y(\(with)17 b(no)g Fp(PROJP1)f Fw(k)o(eyw)o(ord\))i(will)e(b)q(e)i (considered)h(equiv)n(alen)o(t)-116 2307 y(to)e(the)i Fp(AZP)e Fw(pro)r(jection)i(with)e(k)o(eyw)o(ord)h Fp(PROJP1)f Fw(set)i(to)e(zero.)-116 2356 y(Since)11 b(this)f(pro)r(jection)h(is)f (from)f(the)i(cen)o(ter)h(of)e(the)h(sphere,)h(great)-116 2406 y(circles)17 b(are)f(pro)r(jected)i(as)f(straigh)o(t)f(lines.)f (Th)o(us,)i(the)f(shortest)-116 2456 y(distance)f(b)q(et)o(w)o(een)h(t) o(w)o(o)f(p)q(oin)o(ts)f(on)h(the)g(sphere)h(is)f(represen)o(ted)-116 2506 y(as)10 b(a)g(straigh)o(t)g(line)g(in)o(terv)n(al,)f(whic)o(h,)h (ho)o(w)o(ev)o(er,)g(is)g(not)h(uniformly)-116 2556 y(divided.)f(A)h (single)f Fp(TAN)h Fw(pro)r(jection)g(div)o(erges)g(at)g Fo(\022)i Fw(=)f(0,)e(but)i(one)-116 2605 y(ma)o(y)j(use)j(a)f(tangen)o (t-lik)o(e)f(pro)r(jection)i(on)e(six)h(square)h(faces)g(to)-116 2655 y(displa)o(y)12 b(the)j(whole)f(sky)m(.)f(See)h(Section)f(4.4.9)g (for)g(details.)970 1630 y Fm(Fig.)h(4.)g Fe(TAN)d Fs(pro)r(jection,)j (div)o(erges)h(at)e Fk(\022)e Fs(=)g(0)970 1765 y Fw(4.1.3.)h(SIN)i (\(orthographic\))970 1845 y(In)h(radio)f(ap)q(erture)j(syn)o(thesis,)e (the)h(p)q(ersp)q(ectiv)o(e)h(zenithal)e(pro-)970 1894 y(jection)k(whic)o(h)f(arises)i(naturally)d(has)i Fo(\026)g Fw(=)h Fn(1)p Fw(.)e(In)g(this)h(case,)970 1944 y(Eq.)12 b(14)i(b)q(ecomes)970 2058 y Fo(R)1002 2064 y Fi(\022)1041 2058 y Fw(=)1099 2030 y(180)1162 2015 y Fj(\016)p 1099 2048 V 1127 2086 a Fo(\031)1193 2058 y Fw(cos)7 b Fo(\022)1000 2151 y(\022)22 b Fw(=)f(cos)1150 2133 y Fj(\000)p Fu(1)1201 2105 y Fl(\020)1259 2123 y Fo(\031)p 1231 2141 V 1231 2179 a Fw(180)1294 2167 y Fj(\016)1317 2151 y Fo(R)1349 2157 y Fi(\022)1368 2105 y Fl(\021)1406 2151 y Fo(:)970 2257 y Fw(This)9 b(pro)r(jection)h(is)f(illustrated)g(in)g(the)h(upp)q (er)g(p)q(ortion)f(of)g(Fig.)i(5.)970 2307 y(The)22 b Fp(SIN)g Fw(pro)r(jection)g(\(with)f(no)h Fp(PROJP1)f Fw(k)o(eyw)o(ord\))h(will)e(b)q(e)970 2356 y(considered)15 b(equiv)n(alen)o(t)d(to)h(the)h Fp(AZP)f Fw(pro)r(jection)h(with)f(k)o (eyw)o(ord)970 2406 y Fp(PROJP1)d Fw(set)j(to)e(a)g(v)o(ery)g(large)g (n)o(um)o(b)q(er.)g(This)g(pro)r(jection)h(giv)o(es)f(a)970 2456 y(represen)o(tation)17 b(of)d(the)i(visual)e(app)q(earance)i(of)f (a)g(sphere)h(\()p Fr(e.g.)p Fw(,)970 2506 y(a)h(planet\))h(when)g (seen)h(from)c(a)j(great)g(distance.)f(Radio)f(in)o(ter-)970 2556 y(ferometers)21 b(whic)o(h)f(ha)o(v)o(e)g(all)f(baselines)i(orien) o(ted)g(in)f(a)g(plane)970 2605 y(can)d(use)g(a)f(\\slan)o(t)g (orthographic")g(pro)r(jection)h(whic)o(h)f(Greisen)970 2655 y(\(1983)o(\))e(called)f(the)h Fp(NCP)f Fw(pro)r(jection)h(for)f (the)h(particular)g(case)g(of)p eop %%Page: 9 9 9 8 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)448 b(9)-116 731 y Fm(Fig.)14 b(3.)f Fs(Geometry)g(of)g(azim)o(uthal)i(pro) r(jections)-116 862 y Fw(an)k(East-W)m(est)g(in)o(terferometer.)g(A)h (simple)e(extension)i(of)e(the)-116 912 y(form)o(ul\032)12 b(ab)q(o)o(v)o(e,)i(allo)o(ws)g(all)f(suc)o(h)j(cases)g(to)f(b)q(e)g (handled)g(natu-)-116 962 y(rally)m(.)f(W)m(e,)i(therefore,)i(prop)q (ose)f(that)g(the)g Fp(SIN)f Fw(pro)r(jection)h(ac-)-116 1012 y(tually)12 b(b)q(e)j(de\014ned)g(as)-116 1126 y Fo(x)20 b Fw(=)-14 1098 y(180)49 1083 y Fj(\016)p -14 1117 82 2 v 14 1155 a Fo(\031)79 1126 y Fw([)7 b(cos)g Fo(\022)i Fw(sin)e Fo(\036)i Fw(+)g Fo(\013)e Fw(\(sin)g Fo(\022)k Fn(\000)e Fw(1\))e(])293 b(\(18\))-115 1228 y Fo(y)23 b Fw(=)e Fn(\000)18 1200 y Fw(180)81 1185 y Fj(\016)p 18 1219 V 46 1257 a Fo(\031)112 1228 y Fw([)6 b(cos)i Fo(\022)g Fw(cos)g Fo(\036)h Fw(+)g Fo(\014)h Fw(\(sin)d Fo(\022)k Fn(\000)e Fw(1\))e(])13 b Fo(;)232 b Fw(\(19\))-116 1335 y(where)21 b Fo(\013)f Fw(and)h Fo(\014)i Fw(will)c(b)q(e)i(giv)o(en)f(b)o(y)g(k)o(eyw)o(ords)h Fp(PROJP1)e Fw(and)-116 1385 y Fp(PROJP2)p Fw(,)f(resp)q(ectiv)o(ely)m (.)j(F)m(or)e(a)h(planar)g(in)o(terferometer)g(arra)o(y)m(,)-116 1435 y(w)o(e)14 b(ma)o(y)e(write)-116 1530 y Fo(N)-83 1536 y Fi(u)-61 1530 y Fo(u)d Fw(+)g Fo(N)46 1536 y Fi(v)66 1530 y Fo(v)i Fw(+)f Fo(N)172 1536 y Fi(w)199 1530 y Fo(w)i Fw(=)g(0)7 b Fo(;)-116 1625 y Fw(where)17 b(\()p Fo(N)55 1631 y Fi(u)78 1625 y Fo(;)7 b(N)130 1631 y Fi(v)149 1625 y Fo(;)g(N)201 1631 y Fi(w)228 1625 y Fw(\))17 b(are)g(the)g (direction)g(cosines)h(of)e(the)h(nor-)-116 1675 y(mal)c(to)i(the)h (plane)f(in)g(the)h(same)f(co)q(ordinate)h(system)f(used)h(for)-116 1725 y(the)e(\\in)o(terferometer)g(co)q(ordinates")g(\()p Fo(u;)7 b(v)q(;)g(w)q Fw(\).)13 b(Then,)h(the)g(pro-)-116 1775 y(jection)f(parameters)h(are)-116 1870 y Fo(\013)20 b Fw(=)h Fo(N)17 1876 y Fi(u)39 1870 y Fo(=)n(N)91 1876 y Fi(w)-115 1932 y Fo(\014)i Fw(=)e Fo(N)17 1938 y Fi(v)37 1932 y Fo(=)n(N)89 1938 y Fi(w)123 1932 y Fo(:)-116 2028 y Fw(F)m(or)8 b(an)h(East-W)m(est)h(in)o(terferometer,)f(\()p Fo(\013;)e(\014)r Fw(\))12 b(=)g(\(0)p Fo(;)7 b Fw(cot)f Fo(\016)772 2034 y Fu(0)791 2028 y Fw(\),)j(while)-116 2077 y(Corn)o(w)o(ell)14 b(and)h(P)o(erley)g(\(1991\))g(giv)o(e,)f(for) h(an)o(y)f(instan)o(t)h(of)g(time,)-116 2127 y(\()p Fo(\013;)7 b(\014)r Fw(\))18 b(=)g(\(tan)7 b Fo(Z)k Fw(sin)6 b Fo(\037;)h Fw(tan)g Fo(Z)j Fw(cos)d Fo(\037)p Fw(\),)18 b(where)h Fo(Z)i Fw(is)d(the)g(zenith)-116 2177 y(angle)i(and)h Fo(\037)g Fw(is)g(the)h(parallactic)e(angle.)g(Tw)o(o)g(cases)j(of)d (the)-116 2227 y(oblique)15 b(pro)r(jection)i(are)g(illustrated)g(in)f (the)h(lo)o(w)o(er)f(p)q(ortion)g(of)-116 2277 y(Fig.)11 b(5.)-116 2395 y(4.1.4.)g(STG)j(\(stereographic,)h(zenithal)e (orthomorphic\))-116 2475 y(The)18 b(third)g(p)q(ersp)q(ectiv)o(e)j (zenithal)d(pro)r(jection)g(in)g(general)g(use)-116 2525 y(has)c Fo(\026)d Fw(=)h(1,)h(for)h(whic)o(h)f(Eq.)g(14)g(b)q(ecomes) -116 2640 y Fo(R)-84 2646 y Fi(\022)-45 2640 y Fw(=)21 b(2)34 2612 y(180)97 2597 y Fj(\016)p 33 2630 V 62 2668 a Fo(\031)158 2612 y Fw(cos)8 b Fo(\022)p 125 2630 151 2 v 125 2668 a Fw(1)h(+)g(sin)e Fo(\022)849 2640 y Fw(\(20\))1041 881 y(=)21 b(2)1120 853 y(180)1183 838 y Fj(\016)p 1120 872 82 2 v 1148 910 a Fo(\031)1213 881 y Fw(tan)1280 823 y Fl(\022)1316 853 y Fw(90)1358 838 y Fj(\016)1386 853 y Fn(\000)9 b Fo(\022)p 1316 872 132 2 v 1371 910 a Fw(2)1453 823 y Fl(\023)1000 997 y Fo(\022)22 b Fw(=)f(90)1136 980 y Fj(\016)1164 997 y Fn(\000)10 b Fw(2)d(tan)1293 980 y Fj(\000)p Fu(1)1345 939 y Fl(\022)1409 951 y Fi(\031)p 1385 958 67 2 v 1385 981 a Fu(180)1436 973 y Fh(\016)1457 967 y Fo(R)1489 973 y Fi(\022)p 1380 988 128 2 v 1434 1026 a Fw(2)1513 939 y Fl(\023)1557 997 y Fo(:)367 b Fw(\(21\))970 1178 y(This)12 b(pro)r(jection)f(is)h(illustrated)f(in)g (Fig.)h(6.)e(The)i Fp(STG)f Fw(pro)r(jection)970 1228 y(\(with)g(no)f Fp(PROJP1)f Fw(k)o(eyw)o(ord\))i(will)e(b)q(e)j (considered)g(equiv)n(alen)o(t)e(to)970 1278 y(the)15 b Fp(AZP)f Fw(pro)r(jection)h(with)f(k)o(eyw)o(ord)g Fp(PROJP1)f Fw(set)i(to)f(one.)h(This)970 1327 y(pro)r(jection)d(has)g (the)f(prop)q(ert)o(y)i(that)e(all)f(circles)j(on)e(the)h(generat-)970 1377 y(ing)f(sphere)i(are)f(pro)r(jected)g(as)g(circles.)g(It)f(is)g (also)g Fr(orthomorphic)p Fw(,)970 1427 y(meaning)e(that)h(the)h Fo(\036)g Fw(and)f Fo(\022)i Fw(scales)f(are)g(the)g(same)f(at)g(eac)o (h)h(p)q(oin)o(t)970 1477 y(and,)h(therefore,)i(that)f(shap)q(e)h(is)e (preserv)o(ed)j(o)o(v)o(er)e(small)e(regions.)970 2532 y Fm(Fig.)j(6.)g Fe(STG)d Fs(pro)r(jection,)j(div)o(erges)h(at)e Fk(\022)e Fs(=)g Fg(\000)p Fs(90)1704 2516 y Ff(\016)p eop %%Page: 10 10 10 9 bop -116 -192 a Fs(10)430 b(Greisen)14 b(&)e(Calabretta:)i (Represen)o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 1333 y Fm(Fig.)h(5.)f Fe(SIN)f Fs(pro)r(jection,)i(\(top\))f Fk(\013)e Fs(=)h(0)p Fk(;)6 b(\014)14 b Fs(=)e(0,)i(North)g(and)h (South)g(sides)g(b)q(egin)g(to)f(o)o(v)o(erlap)i(at)e Fk(\022)f Fs(=)g(0:)g(\(b)q(ottom)i(left\))f Fk(Z)h Fs(=)d(30)1828 1317 y Ff(\016)1847 1333 y Fk(;)6 b(\037)12 b Fs(=)g(45)1980 1317 y Ff(\016)1999 1333 y Fs(,)-116 1378 y(\(b)q(ottom)h(righ)o(t\))h (East-W)m(est)f(arra)o(y)h(at)f Fk(\016)491 1382 y Ft(0)519 1378 y Fs(=)d(60)597 1363 y Ff(\016)615 1378 y Fs(.)-116 1512 y Fw(4.1.5.)h(AR)o(C)j(\(zenithal)g(equidistan)o(t\))-116 1589 y(There)i(are)g(also)e(some)h(non-p)q(ersp)q(ectiv)o(e)i(zenithal) e(pro)r(jections)-116 1639 y(of)8 b(in)o(terest.)j(The)f(\014rst)g(app) q(eared)h(in)e(Greisen)h(\(1983)o(\))g(as)f Fp(ARC)g Fw(and)-116 1689 y(is)15 b(widely)h(used)h(as)f(the)g(appro)o(ximate)e (pro)r(jection)j(of)e(Sc)o(hmidt)-116 1738 y(telescop)q(es.)f(The)g (nativ)o(e)e(meridians)g(are)h(uniformly)e(divided)h(to)-116 1788 y(giv)o(e)h(uniformly)e(spaced)k(parallels,)e(or)-116 1875 y Fo(R)-84 1881 y Fi(\022)-45 1875 y Fw(=)21 b(\(90)66 1858 y Fj(\016)94 1875 y Fn(\000)9 b Fo(\022)q Fw(\))678 b(\(22\))-87 1937 y Fo(\022)23 b Fw(=)e(90)50 1920 y Fj(\016)77 1937 y Fn(\000)10 b Fo(R)151 1943 y Fi(\022)176 1937 y Fo(:)661 b Fw(\(23\))-116 2025 y(This)9 b(pro)r(jection)i(is)f (illustrated)f(in)h(Fig.)h(7.)e(No)h Fp(PROJP1)f Fw(k)o(eyw)o(ord)-116 2074 y(is)k(required.)-116 2182 y(4.1.6.)e(ZPN)k(\(zenithal)f(p)q (olynomial)o(\))-116 2258 y(An)9 b(extension)h(of)f(the)h Fp(ARC)f Fw(pro)r(jection)h(w)o(ould)e(b)q(e)i(to)f(allo)o(w)f(terms) -116 2308 y(up)13 b(to)h(nin)o(th)g(order)g(in)f(the)i(zenith)f (distance.)g(W)m(e)g(call)f(this)h Fp(ZPN)-116 2358 y Fw(and)f(represen)o(t)j(it)e(as)-116 2482 y Fo(R)-84 2488 y Fi(\022)-55 2482 y Fw(=)-6 2454 y(180)57 2439 y Fj(\016)p -6 2472 82 2 v 22 2511 a Fo(\031)109 2430 y Fu(9)87 2443 y Fl(X)91 2531 y Fi(i)p Fu(=0)154 2482 y Fo(P)181 2488 y Fi(i)202 2436 y Fl(\020)260 2454 y Fo(\031)p 231 2472 V 231 2511 a Fw(180)294 2499 y Fj(\016)318 2482 y Fw(\(90)376 2465 y Fj(\016)404 2482 y Fn(\000)9 b Fo(\022)q Fw(\))481 2436 y Fl(\021)507 2445 y Fi(i)849 2482 y Fw(\(24\))-116 2605 y(where)15 b(the)h Fo(P)104 2611 y Fi(i)132 2605 y Fw(are)f(giv)o(en)g(b)o(y)f(the)i(k)o(eyw)o (ords)f Fp(PROJP0)p Fw(,)e Fp(PROJP1)p Fw(,)-116 2655 y Fo(:)7 b(:)g(:)n Fw(,)13 b Fp(PROJP9)p Fw(,)e(all)h(of)h(whic)o(h)g (ha)o(v)o(e)g(default)g(v)n(alues)g(of)g(zero.)g(Note)970 2276 y Fm(Fig.)h(7.)g Fe(ARC)d Fs(pro)r(jection,)j(no)g(limits)970 2499 y Fw(the)j(units)g(implied)d(b)o(y)i(the)1463 2483 y Fi(\031)p 1440 2490 67 2 v 1440 2514 a Fu(180)1491 2505 y Fh(\016)1512 2499 y Fw(.)g(Since)h(it)f(do)q(es)h(not)f(ha)o(v)o (e)h(an)970 2549 y(exact)i(in)o(v)o(erse,)f(this)g(pro)r(jection)h (should)f(only)f(b)q(e)i(used)g(when)970 2599 y(the)14 b(geometry)f(of)g(the)h(observ)n(ations)g(require)g(it.)f(In)g (particular,)970 2655 y(it)h(should)g(nev)o(er)i(b)q(e)e(used)i(as)e (an)g Fo(n)1542 2640 y Fw(th)1583 2655 y(-order)h(expansion)g(of)e(one) p eop %%Page: 11 11 11 10 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(11)-116 -93 y Fw(of)15 b(the)i(standard)g(zenithal)g(pro)r(jections.) f(An)h(example)e(of)h(this)-116 -43 y(pro)r(jection)e(is)g(illustrated) f(in)h(Fig.)d(8.)-116 852 y Fm(Fig.)22 b(8.)e Fe(ZPN)f Fs(pro)r(jection)i(with)g(parameters,)f(0.05,)g(0.95,)g(-0.025,)-116 897 y(-0.15833,)d(0.00208,)h(0.00792,)g(-0.00007,)f(-0.00019;)h(limits) h(dep)q(end)-116 943 y(up)q(on)14 b(the)f(parameters)-116 1131 y Fw(4.1.7.)e(ZEA)k(\(zenithal)f(equal-area\))-116 1208 y(An)k(equal-area)g(zenithal)h(pro)r(jection,)f(whic)o(h)h(w)o(e)g (will)e(hence-)-116 1258 y(forth)c(call)g Fp(ZEA)p Fw(,)g(ma)o(y)f(b)q (e)j(generated)g(using)-116 1365 y Fo(R)-84 1371 y Fi(\022)-45 1365 y Fw(=)13 1337 y(180)76 1322 y Fj(\016)p 13 1356 82 2 v 41 1394 a Fo(\031)99 1328 y Fl(p)p 140 1328 204 2 v 140 1365 a Fw(2\(1)9 b Fn(\000)h Fw(sin)d Fo(\022)q Fw(\))506 b(\(25\))-45 1470 y(=)21 b(2)34 1442 y(180)97 1427 y Fj(\016)p 33 1461 82 2 v 62 1499 a Fo(\031)127 1470 y Fw(sin)185 1412 y Fl(\022)220 1442 y Fw(90)262 1427 y Fj(\016)290 1442 y Fn(\000)9 b Fo(\022)p 220 1461 132 2 v 276 1499 a Fw(2)357 1412 y Fl(\023)849 1470 y Fw(\(26\))-87 1587 y Fo(\022)23 b Fw(=)e(90)50 1569 y Fj(\016)77 1587 y Fn(\000)10 b Fw(2)d(sin)198 1569 y Fj(\000)p Fu(1)249 1528 y Fl(\022)313 1540 y Fi(\031)p 290 1547 67 2 v 290 1570 a Fu(180)341 1562 y Fh(\016)362 1556 y Fo(R)394 1562 y Fi(\022)p 285 1577 128 2 v 338 1615 a Fw(2)417 1528 y Fl(\023)461 1587 y Fo(:)376 b Fw(\(27\))-116 1697 y(This)9 b(pro)r(jection)i(is)f(illustrated)f(in)h (Fig.)h(9.)e(No)h Fp(PROJP1)f Fw(k)o(eyw)o(ord)-116 1747 y(is)k(required.)-116 1854 y(4.1.8.)e(AIR)j(\(Airy)g(pro)r(jection\)) -116 1931 y(The)i(Airy)g(pro)r(jection)h(is)f(a)f(pro)r(jection)i(whic) o(h)f(minim)o(izes)e(the)-116 1981 y(error)f(for)g(the)h(region)e (within)h(latitude)f Fo(\022)542 1987 y Fi(b)572 1981 y Fw(\(Ev)o(enden,)i(1991)o(\).)f(It)-116 2031 y(is)g(de\014ned)i(b)o (y)-116 2141 y Fo(R)-84 2147 y Fi(\022)-45 2141 y Fw(=)21 b Fn(\000)45 2113 y Fw(180)108 2098 y Fj(\016)p 45 2132 82 2 v 73 2170 a Fo(\031)138 2083 y Fl(\022)174 2113 y Fw(ln)o(\(cos)8 b Fo(\030)r Fw(\))p 174 2132 150 2 v 205 2170 a(tan)f Fo(\030)337 2141 y Fw(+)384 2113 y(ln)o(\(cos)h Fo(\030)515 2119 y Fi(b)532 2113 y Fw(\))p 384 2132 165 2 v 406 2171 a(tan)466 2156 y Fu(2)491 2171 y Fo(\030)509 2177 y Fi(b)560 2141 y Fw(tan)f Fo(\030)647 2083 y Fl(\023)691 2141 y Fo(;)146 b Fw(\(28\))-116 2226 y(where)-86 2304 y Fo(\030)23 b Fn(\021)13 2276 y Fw(90)55 2261 y Fj(\016)82 2276 y Fn(\000)10 b Fo(\022)p 13 2294 132 2 v 68 2332 a Fw(2)-101 2406 y Fo(\030)-83 2412 y Fi(b)-45 2406 y Fn(\021)13 2378 y Fw(90)55 2363 y Fj(\016)82 2378 y Fn(\000)g Fo(\022)143 2384 y Fi(b)p 13 2396 148 2 v 76 2434 a Fw(2)172 2406 y Fo(:)-116 2506 y Fw(When)17 b Fo(\022)27 2512 y Fi(b)61 2506 y Fw(approac)o(hes)h(90)320 2491 y Fj(\016)339 2506 y Fw(,)e(the)i(second)g(term)f(of)g(Eq.)12 b(28)k(ap-)-116 2556 y(proac)o(hes)f(its)f(asymptotic)f(v)n(alue)g(of)h(-1/2.)f(F)m(or) g(all)g Fo(\022)734 2562 y Fi(b)752 2556 y Fw(,)g(the)i(pro-)-116 2605 y(jection)e(is)h(un)o(b)q(ounded)h(at)e(the)i(nativ)o(e)e(south)i (p)q(ole.)e(The)h(in)o(v)o(er-)-116 2655 y(sion)g(of)g(Eq.)e(28)i(ma)o (y)f(not)h(b)q(e)h(expressed)i(so)e(simply)m(,)c(since)16 b(it)e(is)970 701 y Fm(Fig.)g(9.)g Fe(ZEA)d Fs(pro)r(jection,)j(no)g (limits)970 837 y Fw(a)e(transcenden)o(tal)i(equation)e(in)g Fo(\030)j Fw(or)d Fo(\022)q Fw(.)h(One)g(ma)o(y)d(determine)j(a)970 887 y(solution)d(iterativ)o(ely)h(with)f(Newton's)i(metho)q(d)e(\(see)i (Section)h(6\).)970 936 y(P)o(erhaps)h(more)f(simply)m(,)d(one)j(can)g (use)h(Eq.)f(28)f(to)h(\014nd)g(an)g(in)o(ter-)970 986 y(v)n(al)j(in)h Fr(e.g.)p Fw(,)g(cos)7 b Fo(\030)19 b Fw(suc)o(h)f(that)g Fo(R)1506 992 y Fi(\022)1524 995 y Fu(2)1560 986 y Fo(>)f(R)1641 992 y Fi(\022)1677 986 y Fo(>)g(R)1758 992 y Fi(\022)1777 995 y Fu(1)1795 986 y Fw(.)g(Then,)g(one)970 1036 y(iterates)12 b(to)o(w)o(ard)f(the)g (solution)f(alw)o(a)o(ys)g(\014nding)g(an)h(in)o(terv)n(al)f(sur-)970 1086 y(rounding)h(the)h(correct)h(answ)o(er)f(un)o(til)e(con)o(v)o (ergence)j(is)f(ac)o(hiev)o(ed.)970 1136 y(See)j(App)q(endix)e(D.2)g (for)g(an)h(example)e(of)i(this)g(tec)o(hnique.)1032 1188 y(The)e(FITS)f(k)o(eyw)o(ord)g Fp(PROJP1)f Fw(will)g(b)q(e)h(used) h(to)f(sp)q(ecify)h Fo(\022)1947 1194 y Fi(b)1975 1188 y Fw(in)970 1238 y(degrees)h(with)d(a)h(default)g(of)f(90)1459 1222 y Fj(\016)1477 1238 y Fw(.)h(This)g(pro)r(jection)g(is)g (illustrated)970 1287 y(in)j(Fig.)d(10)i(with)h(t)o(w)o(o)f(di\013eren) o(t)i(v)n(alues)f(of)f Fo(\022)1684 1293 y Fi(b)1701 1287 y Fw(.)970 1407 y Fr(4.2.)i(Cylindric)n(al)f(pr)n(oje)n(ctions)970 1487 y Fw(Cylindrical)e(pro)r(jections)i(are)g(a)g(class)f(of)g(pro)r (jections)i(in)e(whic)o(h)970 1537 y(the)19 b(surface)g(of)e(pro)r (jection)h(is)g(a)g(cylinder.)f(The)i(nativ)o(e)e(co)q(or-)970 1587 y(dinate)f(system)g(is)g(c)o(hosen)i(to)e(ha)o(v)o(e)g(its)g(p)q (olar)f(axis)h(coinciden)o(t)970 1637 y(with)f(the)g(axis)f(of)g(the)h (cylinder.)g(Meridians)g(and)f(parallels)g(are)970 1687 y(mapp)q(ed)j(on)o(to)h(a)f(rectangular)h(grid)g(so)g(that)g (cylindrical)f(pro-)970 1736 y(jections)g(are)f(describ)q(ed)i(b)o(y)d (form)o(ul\032)f(whic)o(h)i(return)h Fo(x)e Fw(and)h Fo(y)q Fw(.)970 1786 y(By)10 b(con)o(v)o(en)o(tion,)f(the)i(reference)h (p)q(oin)o(t)d(in)g(all)g(cylindrical)g(pro)r(jec-)970 1836 y(tions)h(is)g(the)h(nativ)o(e)f(co)q(ordinate)g(system)g(origin,) e(\()p Fo(\036;)f(\022)q Fw(\))12 b(=)g(\(0)p Fo(;)7 b Fw(0\).)970 1886 y(F)m(urthermore,)13 b(all)g(cylindrical)g(pro)r (jections)i(ha)o(v)o(e)970 1986 y Fo(x)c Fn(/)h Fo(\036)7 b(:)843 b Fw(\(29\))970 2105 y(4.2.1.)12 b(CYP)i(\(general)g(p)q(ersp)q (ectiv)o(e\))970 2186 y(F)m(or)20 b(general)g(cylindrical)f(p)q(ersp)q (ectiv)o(e,)j(w)o(e)e(pro)r(ject)h(the)g(unit)970 2236 y(sphere)16 b(on)f(a)f(cylinder)h(of)f(radius)h Fo(\025)g Fw(from)e(p)q(oin)o(ts)h(in)g(the)i(equa-)970 2286 y(torial)11 b(plane)h(of)f(the)h(nativ)o(e)g(system)g(at)f(a)h(distance)h Fo(\026)f Fw(measured)970 2336 y(from)f(the)h(cen)o(ter)i(of)d(the)i (sphere)h(in)e(the)g(direction)h(opp)q(osite)f(the)970 2385 y(pro)r(jected)k(surface.)e(It)g(is)g(straigh)o(tforw)o(ard)f(to)h (sho)o(w)g(that)971 2481 y Fo(x)21 b Fw(=)g Fo(\025\036)818 b Fw(\(30\))973 2568 y Fo(y)23 b Fw(=)1074 2540 y(180)1137 2525 y Fj(\016)p 1074 2559 82 2 v 1102 2597 a Fo(\031)1167 2510 y Fl(\022)1232 2540 y Fo(\026)9 b Fw(+)h Fo(\025)p 1202 2559 159 2 v 1202 2597 a(\026)g Fw(+)f(cos)f Fo(\022)1366 2510 y Fl(\023)1404 2568 y Fw(sin)f Fo(\022)h(:)435 b Fw(\(31\))970 2655 y(This)14 b(ma)o(y)e(b)q(e)i(in)o(v)o(erted)h(as)p eop %%Page: 12 12 12 11 bop -116 -192 a Fs(12)430 b(Greisen)14 b(&)e(Calabretta:)i (Represen)o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 793 y Fm(Fig.)i(10.)f Fe(AIR)e Fs(pro)r(jection:)j(\(left\))f Fk(\022)431 797 y Fd(b)460 793 y Fs(=)f(90)542 777 y Ff(\016)575 793 y Fs(\(righ)o(t\))h Fk(\022)717 797 y Fd(b)747 793 y Fs(=)e Fg(\000)p Fs(10)858 777 y Ff(\016)876 793 y Fs(.)i(Both)g(are)g(limited)i(to)d Fk(\022)h(>)e Fg(\000)p Fs(35)1389 777 y Ff(\016)1422 793 y Fs(and)j(div)o(erge)g(at) e Fk(\022)h Fs(=)f Fg(\000)p Fs(90)1826 777 y Ff(\016)1844 793 y Fs(.)g(The)h(t)o(w)o(o)-116 838 y(plots)f(are)f(on)g(the)g(same)g (scale.)-116 975 y Fo(\036)20 b Fw(=)-13 947 y Fo(x)p -13 965 25 2 v -13 1003 a(\025)849 975 y Fw(\(32\))-112 1092 y Fo(\022)i Fw(=)f(arg)7 b(\(1)p Fo(;)g(\021)q Fw(\))i(+)h(sin)243 1075 y Fj(\000)p Fu(1)294 1021 y Fl( )385 1064 y Fo(\021)q(\026)p 332 1083 154 2 v 332 1091 a Fl(p)p 373 1091 113 2 v 373 1128 a Fo(\021)395 1116 y Fu(2)423 1128 y Fw(+)g(1)491 1021 y Fl(!)537 1092 y Fo(;)300 b Fw(\(33\))-116 1211 y(where)-114 1277 y Fo(\021)22 b Fn(\021)15 1249 y Fo(\031)p -13 1268 82 2 v -13 1306 a Fw(180)50 1294 y Fj(\016)117 1249 y Fo(y)p 78 1268 100 2 v 78 1306 a(\026)10 b Fw(+)f Fo(\025)190 1277 y(:)647 b Fw(\(34\))-116 1431 y(Note)10 b(that)h(all)e(v)n(alues)i(of)e Fo(\026)i Fw(are)g(allo)o(w)o(able)d Fr(exc)n(ept)j Fo(\026)h Fw(=)g Fn(\000)p Fo(\025)p Fw(.)e(This)-116 1480 y(pro)r(jection)i(is)h(illustrated)f(in)g(Fig.)f(11.)h(F)m(or)g (FITS)g(purp)q(oses,)i(w)o(e)-116 1530 y(de\014ne)i(the)g(k)o(eyw)o (ord)f Fp(PROJP1)f Fw(to)h(con)o(v)o(ey)g Fo(\026)g Fw(and)g(the)h(k)o (eyw)o(ord)-116 1580 y Fp(PROJP2)e Fw(to)h(con)o(v)o(ey)h Fo(\025)p Fw(.)g(V)m(alues)f(of)g Fo(\026)g(>)f Fw(10)582 1565 y Fu(14)633 1580 y Fw(will)g(b)q(e)i(tak)o(en)g(to)-116 1630 y(b)q(e)d(in\014nite,)g(but)g(if)f Fo(\026)g(>>)h Fw(the)h(n)o(um)o(b)q(er)e(of)h(pixels,)f(then)i(it)f(ma)o(y)-116 1680 y(b)q(e)h(tak)o(en)g(to)g(b)q(e)g(in\014nite)g(with)g(v)o(ery)g (little)f(error.)-116 2648 y Fm(Fig.)h(11.)f Fs(Geometry)h(of)e (cylindrica)q(l)k(pro)r(jections)1032 970 y Fw(Since)c(p)q(ersp)q (ectiv)o(e)h(cylindrical)d(pro)r(jections)i(ha)o(v)o(e)f(not)f(b)q(een) 970 1020 y(in)18 b(wide)g(use)g(in)g(astronom)o(y)m(,)d(w)o(e)j(will)f (de\014ne)i(no)e(sp)q(ecial-case)970 1070 y(pro)r(jections)h(for)f (these)i(as)e(w)o(e)h(did)e(for)h(the)h(zenithal)f(p)q(ersp)q(ec-)970 1120 y(tiv)o(e)24 b(pro)r(jections.)g(W)m(e)g(note)g(that)g(the)g (simplest)f(case)i(puts)970 1169 y(the)20 b(p)q(oin)o(t)f(of)g(pro)r (jection)g(at)g(the)h(cen)o(ter)h(of)e(the)h(sphere)h(and)970 1219 y(uses)g(a)e(unit)h(cylinder)f(\()p Fr(i.e.)p Fw(,)g Fo(\026)i Fw(=)g(0)p Fo(;)7 b(\025)21 b Fw(=)g(1\).)e(In)g(that)h (case,)970 1269 y Fo(y)e Fw(=)1062 1253 y Fu(180)1113 1240 y Fh(\016)p 1062 1260 67 2 v 1085 1283 a Fi(\031)1141 1269 y Fw(tan)7 b Fo(\022)18 b Fw(as)f(illustrated)g(in)f(Fig.)11 b(12.)16 b(Gall's)f(pro)r(jection)970 2203 y Fm(Fig.)25 b(12.)e Fe(CYP)e Fs(pro)r(jection)j(with)f Fk(\026)k Fs(=)f(0)p Fk(;)6 b(\025)27 b Fs(=)f(1,)c(div)o(erges)i(at)970 2248 y Fk(\022)12 b Fs(=)e Fg(\006)p Fs(90)1108 2232 y Ff(\016)970 2371 y Fw(minim)o(izes)17 b(distortions)h(in)g(the)h (equatorial)e(regions)i(with)f(pa-)970 2421 y(rameters)c Fo(\026)e Fw(=)g(1)p Fo(;)7 b(\025)k Fw(=)1340 2386 y Fl(p)p 1382 2386 63 2 v 35 x Fw(1)p Fo(=)p Fw(2)o(,)j(giving)970 2520 y Fo(x)21 b Fw(=)g Fo(\036=)1114 2484 y Fn(p)p 1148 2484 21 2 v 1148 2520 a Fw(2)972 2620 y Fo(y)i Fw(=)1073 2592 y(180)1136 2577 y Fj(\016)p 1073 2610 82 2 v 1101 2648 a Fo(\031)1166 2549 y Fl( )1199 2620 y Fw(1)9 b(+)1275 2558 y Fn(p)p 1310 2558 21 2 v 34 x Fw(2)p 1275 2610 56 2 v 1293 2648 a(2)1336 2549 y Fl(!)1375 2620 y Fw(tan)1442 2561 y Fl(\022)1478 2592 y Fo(\022)p 1478 2610 21 2 v 1478 2648 a Fw(2)1504 2561 y Fl(\023)1548 2620 y Fo(;)p eop %%Page: 13 13 13 12 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(13)-116 -93 y Fw(illustrated)17 b(in)f(Fig.)c(13.)k(Lam)o(b)q(ert's)g (equal-area)h(pro)r(jection)h(is)-116 848 y Fm(Fig.)c(13.)f Fs(Gall's)h Fe(CYP)e Fs(pro)r(jection)i(with)g Fk(\026)d Fs(=)f(1)p Fk(;)c(\025)12 b Fs(=)677 814 y Fl(p)p 719 814 26 2 v 724 833 a Ft(1)p 724 840 16 2 v 724 861 a(2)744 848 y Fs(,)g(no)i(limits)-116 990 y Fw(a)f(sp)q(ecial)g(case)h(of)f(b)q (oth)h(the)f(general)h(p)q(ersp)q(ectiv)o(e)i(and)d(general)-116 1039 y(equal-area)h(cylindrical)h(pro)r(jections.)g(It)g(has)h (parameters)f Fo(\026)f Fw(=)-116 1089 y Fn(1)p Fo(;)7 b(\025)j Fw(=)i(1,)h(equations)-116 1184 y Fo(x)20 b Fw(=)h Fo(\036)-115 1265 y(y)i Fw(=)-14 1237 y(180)49 1222 y Fj(\016)p -14 1255 82 2 v 14 1293 a Fo(\031)79 1265 y Fw(sin)7 b Fo(\022)i(;)-116 1371 y Fw(and)k(is)h(illustrated)g (in)f(Fig.)e(14.)-116 1487 y(4.2.2.)g(CAR)j(\(Cartesian\))-116 1566 y(The)19 b(Cartesian,)h(or)f(Plate)h(Carr)o(\023)-20 b(ee,)20 b(pro)r(jection)g(is)f(the)h(abso-)-116 1616 y(lutely)13 b(simplest)h(pro)r(jection)g(one)h(could)f(prop)q(ose.)h (Its)f(form)o(ul\032)-116 1666 y(are)-116 1760 y Fo(x)20 b Fw(=)h Fo(\036)843 b Fw(\(35\))-115 1823 y Fo(y)23 b Fw(=)e Fo(\022)9 b(:)828 b Fw(\(36\))-116 1917 y(This)21 b(pro)r(jection)i(is)f(illustrated)f(in)h(Fig.)11 b(15.)21 b(The)i(k)o(eyw)o(ords)-116 1967 y Fp(PROJP1)12 b Fw(and)i Fp(PROJP2)f Fw(are)i(not)f(used.)g(If)g(no)g(pro)r(jection)g(t)o(yp)q (e)h(is)-116 2017 y(sp)q(eci\014ed,)f(then)f(FITS)g(readers)i(should)e Fr(not)g Fw(assume)g(the)g(linear)-116 2067 y(Cartesian)i Fp(CAR)p Fw(.)g(They)g(m)o(ust)g(assume,)f(instead,)h(simple)f(linear) -116 2116 y(co)q(ordinates)h(with)g(none)h(of)e(the)i(celestial-co)q (ordinate)g(con)o(v)o(en-)-116 2166 y(tions)d(of)h(this)f(man)o (uscript.)-116 2282 y(4.2.3.)e(MER)j(\(Mercator\))-116 2362 y(The)g(Mercator)h(pro)r(jection)f(has)h(b)q(een)g(widely)e(used)i (in)e(na)o(viga-)-116 2411 y(tion)c(since)i(it)e(has)i(the)f(prop)q (ert)o(y)h(that)f(lines)g(of)f(constan)o(t)i(b)q(earing)-116 2461 y(\()p Fr(rhumb)k(lines)p Fw(\))e(are)i(pro)r(jected)g(as)f (straigh)o(t)g(lines.)g(Since)g(this)g(is)-116 2511 y(the)e(result)h (of)e(a)h(di\013eren)o(tial)g(equation,)f(this)h(has)g(the)g(nast)o(y)g (for-)-116 2561 y(m)o(ulation)-116 2655 y Fo(x)20 b Fw(=)h Fo(\036)843 b Fw(\(37\))972 -74 y Fo(y)23 b Fw(=)1073 -102 y(180)1136 -117 y Fj(\016)p 1073 -84 V 1101 -46 a Fo(\031)1166 -74 y Fw(ln)6 b(tan)1274 -133 y Fl(\022)1310 -102 y Fw(90)1352 -117 y Fj(\016)1380 -102 y Fw(+)j Fo(\022)p 1310 -84 132 2 v 1365 -46 a Fw(2)1447 -133 y Fl(\023)1491 -74 y Fo(;)433 b Fw(\(38\))970 11 y(whic)o(h)14 b(rev)o(erses)i(to)973 77 y Fo(\022)23 b Fw(=)e(2)7 b(tan)1155 60 y Fj(\000)p Fu(1)1207 31 y Fl(\020)1232 77 y Fo(e)1251 60 y Fi(y)q(\031)q(=)p Fu(180)1357 47 y Fh(\016)1375 31 y Fl(\021)1409 77 y Fn(\000)j Fw(90)1493 60 y Fj(\016)1518 77 y Fo(:)406 b Fw(\(39\))970 174 y(This)22 b(pro)r(jection)g(is)g(illustrated)g(in)f (Fig.)12 b(16.)21 b(The)h(k)o(eyw)o(ords)970 224 y Fp(PROJP1)13 b Fw(and)g Fp(PROJP2)g Fw(are)h(not)g(used.)970 1142 y Fm(Fig.)g(16.)g Fe(MER)d Fs(pro)r(jection,)j(div)o(erges)h(at)e Fk(\022)f Fs(=)e Fg(\006)p Fs(90)1726 1126 y Ff(\016)970 1338 y Fw(4.2.4.)i(CEA)i(\(equal)g(area\))970 1414 y(P)o(arallels)k (are)h(spaced)h(so)e(as)h(to)g(mak)o(e)e(the)i(pro)r(jection)g(equal) 970 1464 y(area)g(and)f(scaled)g(so)h(as)f(to)g(mak)o(e)f(it)g (conformal)f(at)j(latitudes)970 1514 y Fo(\022)13 b Fw(=)f Fn(\006)p Fo(\022)1097 1520 y Fi(x)1119 1514 y Fw(.)h(De\014ning)h Fo(\025)d Fn(\021)h Fw(cos)1445 1499 y Fu(2)1470 1514 y Fo(\022)1489 1520 y Fi(x)1511 1514 y Fw(,)h(the)i(form)o(ul\032)c (are)970 1601 y Fo(x)21 b Fw(=)g Fo(\036)843 b Fw(\(40\))972 1685 y Fo(y)23 b Fw(=)1073 1657 y(180)1136 1642 y Fj(\016)p 1073 1675 82 2 v 1101 1713 a Fo(\031)1164 1657 y Fw(sin)7 b Fo(\022)p 1164 1675 79 2 v 1191 1713 a(\025)1254 1685 y(;)670 b Fw(\(41\))970 1759 y(whic)o(h)14 b(rev)o(erses)i(to)973 1825 y Fo(\022)23 b Fw(=)e(sin)1119 1808 y Fj(\000)p Fu(1)1170 1779 y Fl(\020)1228 1797 y Fo(\031)p 1200 1815 82 2 v 1200 1853 a Fw(180)1263 1841 y Fj(\016)1286 1825 y Fo(y)q(\025)1331 1779 y Fl(\021)1371 1825 y Fo(:)553 b Fw(\(42\))970 1924 y(The)13 b(k)o(eyw)o(ords)f Fp(PROJP1)f Fw(is)h(used)g(to)g(giv)o(e)g Fo(\025)g Fw(whic)o(h)g(m)o(ust)f (satisfy)970 1973 y(0)16 b Fo(<)h(\025)f Fn(\024)g Fw(1.)g(Lam)o(b)q (ert's)g(pro)r(jection)h(is)g(the)g(case)h(with)e Fo(\025)g Fw(=)h(1)970 2023 y(\(see)e(Fig.)d(14\).)970 2130 y Fr(4.3.)j(Conic)g (pr)n(oje)n(ctions)970 2207 y Fw(Conic)h(pro)r(jections)h(are)g(a)f (class)g(of)g(pro)r(jections)h(in)f(whic)o(h)g(the)970 2257 y(surface)f(of)e(pro)r(jection)i(is)f(a)f(cone)i(and)e(parallels)h (are)g(pro)r(jected)970 2307 y(as)f(arcs)h(of)e(circles.)h(In)g (\\standard")g(conic)g(pro)r(jections)h(the)f(axis)970 2356 y(of)18 b(the)i(cone)f(is)f(coinciden)o(t)h(with)g(the)g(p)q(olar) f(axis)h(of)f(the)h(na-)970 2406 y(tiv)o(e)13 b(co)q(ordinate)f (system,)g(allo)o(wing)e(them)i(to)h(b)q(e)g(c)o(haracterized)970 2456 y(b)o(y)h(the)g(t)o(w)o(o)f(\(nativ)o(e\))h(latitudes)g(at)f(whic) o(h)h(the)g(cone)g(in)o(tersects)970 2506 y(the)i(unit)e(sphere.)j(In)d (\\one-standard")h(conic)h(pro)r(jections,)f(the)970 2556 y(cone)j(is)f(exactly)g(tangen)o(t)g(to)g(the)h(sphere)h(and)e (the)g(t)o(w)o(o)g(para-)970 2605 y(metric)h(latitudes)h(b)q(ecome)f (equal.)g(\(See)i(the)f(top)f(p)q(ortions)h(of)970 2655 y(Fig.)12 b(17.\))k(In)g(all)g(of)g(the)i(conic)f(pro)r(jections)g (presen)o(ted)i(b)q(elo)o(w,)p eop %%Page: 14 14 14 13 bop -116 -192 a Fs(14)430 b(Greisen)14 b(&)e(Calabretta:)i (Represen)o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 701 y Fm(Fig.)g(14.)f Fs(Lam)o(b)q(ert's)g(equal)i(area)e Fe(CYP)e Fs(pro)r(jection)k(with)e Fk(\026)e Fs(=)g Fg(1)p Fk(;)6 b(\025)11 b Fs(=)g(1,)h(no)i(limits)-116 1593 y Fm(Fig.)g(15.)f Fe(CAR)f Fs(pro)r(jection,)i(no)f(limits)-116 1725 y Fw(the)i(parallels)f(are)h(concen)o(tric)h(and)e(ma)o(y)f(b)q(e) i(describ)q(ed)i(b)o(y)d Fo(R)893 1731 y Fi(\022)911 1725 y Fw(,)-116 1774 y(the)h(radius)g(for)f(latitude)g Fo(\022)q Fw(,)h(and)f Fo(A)460 1780 y Fi(\036)483 1774 y Fw(,)g(the)h(angle)f(for)h(longitude)-116 1824 y Fo(\036)p Fw(,)e(but)i(an)g(o\013set)g(in)g Fo(y)h Fw(is)f(required)g(to)g(force) g(\()p Fo(x;)7 b(y)q Fw(\))13 b(=)g(\(0)p Fo(;)7 b Fw(0\))14 b(at)-116 1874 y(the)g(reference)i(p)q(oin)o(t.)d(Then)-116 1952 y Fo(x)20 b Fw(=)h Fo(R)13 1958 y Fi(\022)38 1952 y Fw(sin)7 b Fo(A)127 1958 y Fi(\036)849 1952 y Fw(\(43\))-115 2014 y Fo(y)23 b Fw(=)e Fo(Y)5 2020 y Fu(0)33 2014 y Fn(\000)10 b Fo(R)107 2020 y Fi(\022)132 2014 y Fw(cos)d Fo(A)225 2020 y Fi(\036)254 2014 y Fo(:)583 b Fw(\(44\))-116 2091 y(By)18 b(con)o(v)o(en)o(tion,)e(the)j(reference)h(p)q(oin)o(t)d (in)g(all)g(conical)g(pro)r(jec-)-116 2141 y(tions)23 b(is)h(the)g(nativ)o(e)f(co)q(ordinate)h(system)g(cen)o(tral)g (latitude,)-116 2191 y(\()p Fo(\036;)7 b(\022)q Fw(\))k(=)h(\(0)p Fo(;)7 b(\022)110 2197 y Fi(a)130 2191 y Fw(\),)12 b(where)i Fo(\022)308 2197 y Fi(a)340 2191 y Fn(\021)e Fw(\()p Fo(\022)419 2197 y Fu(1)444 2191 y Fw(+)7 b Fo(\022)502 2197 y Fu(2)521 2191 y Fw(\))p Fo(=)p Fw(2.)12 b(None)h(of)f(the)h (equa-)-116 2241 y(tions)i(b)q(elo)o(w)g(require)i Fo(\022)267 2247 y Fu(2)300 2241 y Fo(>)e(\022)366 2247 y Fu(1)401 2241 y Fw(and,)g(if)g(w)o(e)h(allo)o(w)e(the)i(general-)-116 2291 y(ization)-114 2368 y Fo(R)-82 2374 y Fi(\022)-43 2368 y Fw(=)22 b(sign)o Fo(\022)101 2374 y Fi(a)122 2331 y Fl(p)p 163 2331 260 2 v 163 2368 a Fo(x)187 2356 y Fu(2)215 2368 y Fw(+)9 b(\()p Fo(Y)296 2374 y Fu(0)325 2368 y Fn(\000)g Fo(y)q Fw(\))403 2356 y Fu(2)849 2368 y Fw(\(45\))-116 2455 y Fo(A)-85 2461 y Fi(\036)-43 2455 y Fw(=)22 b(arg)83 2397 y Fl(\022)118 2427 y Fo(Y)142 2433 y Fu(0)170 2427 y Fn(\000)10 b Fo(y)p 118 2446 116 2 v 151 2484 a(R)183 2490 y Fi(\022)239 2455 y Fo(;)275 2427 y(x)p 262 2446 51 2 v 262 2484 a(R)294 2490 y Fi(\022)317 2397 y Fl(\023)362 2455 y Fo(:)475 b Fw(\(46\))-116 2556 y(then)14 b(none)g(require)h(that)f Fo(\022)326 2562 y Fu(1)357 2556 y Fn(\025)d Fw(0.)-54 2605 y(One)j(should)f(note)h (that)f(the)h(zenithal)g(pro)r(jections)g(are)g(just)-116 2655 y(sp)q(ecial)i(cases)i(of)e(the)h(conicals)g(\(with)f Fo(\022)542 2661 y Fu(1)577 2655 y Fw(=)g Fo(\022)644 2661 y Fu(2)679 2655 y Fw(=)h(90)770 2640 y Fj(\016)788 2655 y Fw(\).)f(Lik)o(e-)970 1725 y(wise,)e(the)g(cylindrical)f(pro)r (jections)h(corresp)q(ond)i(to)d(conicals)h(in)970 1774 y(the)h(limit)d Fo(\022)1161 1780 y Fu(1)1193 1774 y Fw(=)h Fo(\022)1257 1780 y Fu(2)1289 1774 y Fw(=)g(0.)h(W)m(e)g(presen) o(t)i(the)f(zenithal)g(and)f(cylin-)970 1824 y(drical)9 b(pro)r(jections)i(separately)f(b)q(ecause)h(of)e(their)h(general)g (usage)970 1874 y(and)g(b)q(ecause)i(of)e(the)h(mathematical)c (complexit)o(y)h(of)i(the)h(conical)970 1924 y(pro)r(jection)17 b(form)o(ul\032)d(in)i(these)i(limits.)13 b(W)m(e)j(regard)h(it)f(as)g (p)q(o)q(or)970 1974 y(form)c(to)i(describ)q(e)i(pro)r(jections)e(as)g (conicals)g(at)g(these)h(limits.)970 2087 y(4.3.1.)d(COP)i(\(p)q(ersp)q (ectiv)o(e\))970 2166 y(The)i(t)o(w)o(o-standard)f(p)q(ersp)q(ectiv)o (e)j(pro)r(jection)d(is)h(c)o(haracterized)970 2215 y(b)o(y)j(t)o(w)o (o)g(angles,)f Fn(\000)p Fw(90)1332 2200 y Fj(\016)1371 2215 y Fn(\024)i Fo(\022)1442 2221 y Fu(1)1482 2215 y Fn(\024)g Fo(\022)1553 2221 y Fu(2)1592 2215 y Fn(\024)h Fw(90)1687 2200 y Fj(\016)1705 2215 y Fw(.)e(The)g(pro)r(jection)970 2265 y(form)o(ul\032)12 b(are)973 2370 y Fo(R)1005 2376 y Fi(\022)1044 2370 y Fw(=)1102 2342 y(180)1165 2326 y Fj(\016)p 1102 2360 82 2 v 1130 2398 a Fo(\031)1195 2370 y Fw(cos)c Fo(\013)f Fw([cot)f Fo(\022)1384 2376 y Fi(a)1414 2370 y Fn(\000)k Fw(tan\()p Fo(\022)h Fn(\000)e Fo(\022)1622 2376 y Fi(a)1643 2370 y Fw(\)])265 b(\(47\))1044 2472 y(=)1102 2444 y(180)1165 2428 y Fj(\016)p 1102 2462 V 1130 2500 a Fo(\031)1255 2444 y Fw(cos)8 b Fo(\013)f Fw(cos)g Fo(\022)p 1193 2462 304 2 v 1193 2500 a Fw(sin)g Fo(\022)1270 2506 y Fi(a)1298 2500 y Fw(cos\()p Fo(\022)k Fn(\000)f Fo(\022)1460 2506 y Fi(a)1480 2500 y Fw(\))970 2558 y Fo(A)1001 2564 y Fi(\036)1044 2558 y Fw(=)21 b Fo(\036)7 b Fw(sin)g Fo(\022)1206 2564 y Fi(a)1936 2558 y Fw(\(48\))981 2643 y Fo(Y)1005 2649 y Fu(0)1044 2643 y Fw(=)1102 2615 y(180)1165 2600 y Fj(\016)p 1102 2634 82 2 v 1130 2672 a Fo(\031)1195 2643 y Fw(cos)h Fo(\013)f Fw(cot)g Fo(\022)1373 2649 y Fi(a)1400 2643 y Fo(;)524 b Fw(\(49\))p eop %%Page: 15 15 15 14 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(15)-116 -93 y Fw(where)-103 -26 y Fo(\022)-84 -20 y Fi(a)-43 -26 y Fn(\021)22 b Fw(\()p Fo(\022)46 -20 y Fu(1)74 -26 y Fw(+)10 b Fo(\022)135 -20 y Fu(2)154 -26 y Fw(\))p Fo(=)p Fw(2)637 b(\(50\))-90 36 y Fo(\013)20 b Fn(\021)i Fw(\()p Fo(\022)46 42 y Fu(2)74 36 y Fn(\000)10 b Fo(\022)135 42 y Fu(1)154 36 y Fw(\))p Fo(=)p Fw(2)d Fo(:)618 b Fw(\(51\))-116 98 y(The)14 b(in)o(v)o(erse)g(ma)o(y)e(b)q(e) j(computed)e(using)h(Eqs.)e(45)i(and)g(46)f(with)-88 183 y Fo(\036)20 b Fw(=)38 155 y Fo(A)69 161 y Fi(\036)p 16 174 98 2 v 16 212 a Fw(sin)6 b Fo(\022)92 218 y Fi(a)849 183 y Fw(\(52\))-84 295 y Fo(\022)22 b Fw(=)g Fo(\022)30 301 y Fi(a)59 295 y Fw(+)10 b(tan)161 278 y Fj(\000)p Fu(1)212 236 y Fl(\022)243 295 y Fw(cot)d Fo(\022)324 301 y Fi(a)354 295 y Fn(\000)428 267 y Fo(\031)p 400 285 82 2 v 400 323 a Fw(180)463 311 y Fj(\016)511 267 y Fo(R)543 273 y Fi(\022)p 492 285 90 2 v 492 323 a Fw(cos)g Fo(\013)586 236 y Fl(\023)630 295 y Fo(:)207 b Fw(\(53\))-116 396 y(This)13 b(pro)r(jection)h(is)f(illustrated)g(in)g(Fig.)f(17)h (with)g(t)o(w)o(o)g(di\013eren)o(t)-116 446 y(parameterizations.)k(The) h(k)o(eyw)o(ord)g Fp(PROJP1)f Fw(will)g(b)q(e)i(used)g(to)-116 496 y(giv)o(e)d Fo(\022)-9 502 y Fi(a)28 496 y Fw(in)h(degrees)h(and)f (the)h(k)o(eyw)o(ord)e Fp(PROJP2)g Fw(will)f(b)q(e)j(used)-116 546 y(to)g(giv)o(e)h Fo(\013)f Fw(in)g(degrees.)j(The)e(default)f(v)n (alue)g(for)h Fp(PROJP2)e Fw(is)i(0)-116 596 y(while)11 b Fp(PROJP1)f Fw(has)i(no)f(default.)g(It)h(is)f(recommended)g(that)h (b)q(oth)-116 646 y(k)o(eyw)o(ords)21 b(alw)o(a)o(ys)e(b)q(e)i(giv)o (en.)f(The)h(case)h Fo(\022)606 652 y Fu(1)648 646 y Fw(=)h Fn(\000)p Fo(\022)754 652 y Fu(2)794 646 y Fw(giv)o(es)d(a)-116 695 y(div)o(ergen)o(t)f(equation)f(for)h Fo(R)345 701 y Fi(\022)363 695 y Fw(,)g(but)g(not)g(for)f(\()p Fo(x;)7 b(y)q Fw(\).)19 b(This)g(case)-116 745 y(should)13 b(b)q(e)i(written)f (as)g(a)g(cylindrical)f(pro)r(jection)h(\(whic)o(h)g(it)g(is\))-116 795 y(rather)20 b(than)f(a)h(conical)f(one)g(in)g(order)i(to)e(a)o(v)o (oid)f(p)q(osing)h(an)o(y)-116 845 y(mathematical)12 b(di\016culties.)k(T)m(o)f(do)h(otherwise)h(is,)f(w)o(e)g(b)q(eliev)o 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Fo(A)69 2385 y Fi(\036)p 16 2397 98 2 v 16 2435 a Fw(sin)6 b Fo(\022)92 2441 y Fi(a)849 2407 y Fw(\(64\))-84 2515 y Fo(\022)22 b Fw(=)g Fo(\022)30 2521 y Fi(a)59 2515 y Fw(+)106 2487 y(180)169 2472 y Fj(\016)p 106 2505 82 2 v 134 2543 a Fo(\031)199 2515 y Fw(cot)7 b Fo(\022)280 2521 y Fi(a)310 2515 y Fn(\000)j Fo(R)384 2521 y Fi(\022)409 2515 y Fo(:)428 b Fw(\(65\))-116 2605 y(This)13 b(pro)r(jection)h(is)f(illustrated)g (in)g(Fig.)f(18)h(with)g(t)o(w)o(o)g(di\013eren)o(t)-116 2655 y(parameterizations.)k(The)h(k)o(eyw)o(ord)g Fp(PROJP1)f Fw(will)g(b)q(e)i(used)g(to)970 -93 y(giv)o(e)e Fo(\022)1078 -87 y Fi(a)1115 -93 y Fw(in)f(degrees)j(and)e(the)g(k)o(eyw)o(ord)g Fp(PROJP2)f Fw(will)f(b)q(e)j(used)970 -43 y(to)h(giv)o(e)f Fo(\013)h Fw(in)f(degrees.)i(The)f(default)g(v)n(alue)f(for)h Fp(PROJP2)e Fw(is)i(0)970 7 y(while)12 b Fp(PROJP1)e Fw(has)j(no)e(default;)h(it)f(is)h(recommended)f(that)h(b)q(oth)970 57 y(k)o(eyw)o(ords)i(alw)o(a)o(ys)f(b)q(e)i(giv)o(en.)1032 106 y(The)j(Murdo)q(c)o(h)h(conical)e(pro)r(jection)h(\(P)o(earson,)g (1984)o(\))f(has)970 156 y(the)e(form)973 256 y Fo(R)1005 262 y Fi(\022)1044 256 y Fw(=)21 b Fo(\022)1116 262 y Fi(a)1146 256 y Fn(\000)9 b Fo(\022)i Fw(+)1259 198 y Fl(\022)1294 228 y Fw(180)1357 213 y Fj(\016)p 1294 247 V 1322 285 a Fo(\031)1381 198 y Fl(\023)1411 207 y Fu(2)1442 228 y Fw(sin)c Fo(\013)p 1442 247 85 2 v 1471 285 a(\013)1538 256 y Fw(cos)h Fo(\022)1620 262 y Fi(a)970 341 y Fo(A)1001 347 y Fi(\036)1044 341 y Fw(=)21 b Fo(\036)7 b Fw(sin)g Fo(\022)1206 347 y Fi(a)981 438 y Fo(Y)1005 444 y Fu(0)1044 438 y Fw(=)1097 380 y Fl(\022)1133 410 y Fw(180)1196 395 y Fj(\016)p 1133 429 82 2 v 1161 467 a Fo(\031)1219 380 y Fl(\023)1250 389 y Fu(2)1280 410 y Fw(sin)g Fo(\013)p 1280 429 85 2 v 1309 467 a(\013)1377 438 y Fw(cos)g Fo(\022)1458 444 y Fi(a)1486 438 y Fo(;)970 538 y Fw(whic)o(h)j(is)h(so)f(similar)e (to)i(the)h(conical)f(equidistan)o(t)g(that)h(w)o(e)f(reject)970 588 y(it)k(as)g(a)f(separate)i(FITS-standard)g(pro)r(jection.)970 694 y(4.3.3.)d(COE)i(\(equal)g(area\))970 770 y(The)19 b(t)o(w)o(o-standard)e(equal)h(area)g(pro)r(jection)h(is)f(c)o (haracterized)970 820 y(b)o(y)h(t)o(w)o(o)g(angles,)f Fn(\000)p Fw(90)1332 805 y Fj(\016)1371 820 y Fn(\024)i Fo(\022)1442 826 y Fu(1)1482 820 y Fn(\024)g Fo(\022)1553 826 y Fu(2)1592 820 y Fn(\024)h Fw(90)1687 805 y Fj(\016)1705 820 y Fw(.)e(The)g(pro)r(jection)970 870 y(form)o(ul\032)12 b(are)973 958 y Fo(R)1005 964 y Fi(\022)1044 958 y Fw(=)1102 930 y(180)1165 915 y Fj(\016)p 1102 948 82 2 v 1130 986 a Fo(\031)1195 930 y Fw(2)p 1193 948 24 2 v 1193 986 a Fo(\015)1222 920 y Fl(p)p 1264 920 431 2 v 38 x Fw(1)d(+)g(sin)e Fo(\022)1412 964 y Fu(1)1438 958 y Fw(sin)g Fo(\022)1515 964 y Fu(2)1543 958 y Fn(\000)j Fo(\015)g Fw(sin)c Fo(\022)244 b Fw(\(66\))970 1042 y Fo(A)1001 1048 y Fi(\036)1044 1042 y Fw(=)21 b Fo(\036\015)r(=)p Fw(2)749 b(\(67\))981 1127 y Fo(Y)1005 1133 y Fu(0)1044 1127 y Fw(=)1102 1099 y(180)1165 1084 y Fj(\016)p 1102 1118 82 2 v 1130 1156 a Fo(\031)1195 1099 y Fw(2)p 1193 1118 24 2 v 1193 1156 a Fo(\015)1222 1090 y Fl(p)p 1264 1090 636 2 v 37 x Fw(1)9 b(+)g(sin)e Fo(\022)1412 1133 y Fu(1)1438 1127 y Fw(sin)g Fo(\022)1515 1133 y Fu(2)1543 1127 y Fn(\000)j Fo(\015)g Fw(sin\(\()p Fo(\022)1718 1133 y Fu(1)1746 1127 y Fw(+)g Fo(\022)1807 1133 y Fu(2)1826 1127 y Fw(\))p Fo(=)p Fw(2\))d Fo(;)17 b Fw(\(68\))970 1209 y(where)1000 1276 y Fo(\015)23 b Fn(\021)e Fw(sin)7 b Fo(\022)1174 1282 y Fu(1)1202 1276 y Fw(+)j(sin)d Fo(\022)1321 1282 y Fu(2)1347 1276 y Fo(;)577 b Fw(\(69\))970 1338 y(or,)14 b(using)f(the)i(a)o(v)o(erage) f(and)f(di\013erence)j(angles,)973 1438 y Fo(R)1005 1444 y Fi(\022)1044 1438 y Fw(=)1102 1410 y(180)1165 1395 y Fj(\016)p 1102 1429 82 2 v 1130 1467 a Fo(\031)1193 1370 y Fl(p)p 1235 1370 633 2 v 40 x Fw(cos)1291 1398 y Fu(2)1316 1410 y Fo(\013)9 b Fw(+)h(sin)1444 1393 y Fu(2)1470 1410 y Fo(\022)1489 1416 y Fi(a)1519 1410 y Fn(\000)f Fw(2)e(sin)g Fo(\022)1665 1416 y Fi(a)1692 1410 y Fw(cos)h Fo(\013)e Fw(sin)h Fo(\022)p 1193 1429 674 2 v 1433 1467 a Fw(sin)g Fo(\022)1510 1473 y Fi(a)1538 1467 y Fw(cos)g Fo(\013)1936 1438 y Fw(\(70\))970 1521 y Fo(A)1001 1527 y Fi(\036)1044 1521 y Fw(=)21 b Fo(\036)7 b Fw(sin)g Fo(\022)1206 1527 y Fi(a)1233 1521 y Fw(cos)h Fo(\013)613 b Fw(\(71\))981 1642 y Fo(Y)1005 1648 y Fu(0)1044 1642 y Fw(=)1102 1614 y(180)1165 1599 y Fj(\016)p 1102 1632 82 2 v 1130 1670 a Fo(\031)1193 1553 y Fl(q)p 1235 1553 496 2 v 50 x Fw(cos)1291 1591 y Fu(2)1316 1603 y Fo(\013)9 b Fw(+)h(sin)1444 1586 y Fu(2)1470 1603 y Fo(\022)1489 1609 y Fi(a)1510 1603 y Fw(\(1)f Fn(\000)g Fw(2)e(cos)g Fo(\013)p Fw(\))p 1193 1632 537 2 v 1365 1670 a(sin)g Fo(\022)1442 1676 y Fi(a)1469 1670 y Fw(cos)h Fo(\013)1936 1642 y Fw(\(72\))970 1722 y(where)984 1788 y Fo(\022)1003 1794 y Fi(a)1044 1788 y Fn(\021)21 b Fw(\()p Fo(\022)1132 1794 y Fu(1)1161 1788 y Fw(+)9 b Fo(\022)1221 1794 y Fu(2)1240 1788 y Fw(\))p Fo(=)p Fw(2)638 b(\(73\))997 1851 y Fo(\013)20 b Fn(\021)h Fw(\()p Fo(\022)1132 1857 y Fu(2)1161 1851 y Fn(\000)9 b Fo(\022)1221 1857 y Fu(1)1240 1851 y Fw(\))p Fo(=)p Fw(2)e Fo(:)619 b Fw(\(74\))970 1928 y(Using)14 b(Eqs.)e(45)i(and)f(46,)g(the)i(in)o(v)o(erse)f (equations)g(b)q(ecome)970 2022 y Fo(\036)21 b Fw(=)1075 1994 y(2)p 1074 2013 24 2 v 1074 2051 a Fo(\015)1102 2022 y(A)1133 2028 y Fi(\036)1936 2022 y Fw(\(75\))974 2136 y Fo(\022)i Fw(=)e(sin)1120 2118 y Fj(\000)p Fu(1)1171 2077 y Fl(\022)1208 2108 y Fw(1)p 1207 2126 V 1207 2164 a Fo(\015)1245 2136 y Fw(+)1291 2108 y(sin)7 b Fo(\022)1368 2114 y Fu(1)1394 2108 y Fw(sin)g Fo(\022)1471 2114 y Fu(2)p 1291 2126 199 2 v 1379 2164 a Fo(\015)1130 2252 y Fn(\000)j Fo(\015)r(R)1227 2235 y Fu(2)1227 2262 y Fi(\022)1260 2206 y Fl(\020)1318 2224 y Fo(\031)p 1290 2242 82 2 v 1290 2280 a Fw(360)1353 2268 y Fj(\016)1376 2206 y Fl(\021)1401 2214 y Fu(2)1419 2193 y Fl(\023)1464 2252 y Fo(;)460 b Fw(\(76\))970 2326 y(or)970 2403 y Fo(\036)21 b Fw(=)1144 2375 y Fo(A)1175 2381 y Fi(\036)p 1074 2393 194 2 v 1074 2431 a Fw(sin)7 b Fo(\022)1151 2437 y Fi(a)1178 2431 y Fw(cos)h Fo(\013)1936 2403 y Fw(\(77\))974 2514 y Fo(\022)23 b Fw(=)e(sin)1120 2497 y Fj(\000)p Fu(1)1171 2456 y Fl(\022)1225 2486 y Fw(cos)7 b Fo(\013)p 1207 2505 126 2 v 1207 2543 a Fw(2)g(sin)f Fo(\022)1311 2549 y Fi(a)1346 2514 y Fw(+)1402 2486 y(sin)h Fo(\022)1479 2492 y Fi(a)p 1393 2505 117 2 v 1393 2543 a Fw(2)g(cos)g Fo(\013)1128 2632 y Fn(\000)1165 2604 y Fw(sin)g Fo(\022)1242 2610 y Fi(a)1269 2604 y Fw(cos)h Fo(\013R)1391 2589 y Fu(2)1391 2616 y Fi(\022)p 1165 2623 245 2 v 1277 2661 a Fw(2)1421 2586 y Fl(\020)1479 2604 y Fo(\031)p 1451 2623 82 2 v 1451 2661 a Fw(180)1514 2649 y Fj(\016)1537 2586 y Fl(\021)1562 2595 y Fu(2)1581 2574 y Fl(\023)1625 2632 y Fo(:)299 b Fw(\(78\))p eop %%Page: 16 16 16 15 bop -116 -192 a Fs(16)430 b(Greisen)14 b(&)e(Calabretta:)i (Represen)o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 1421 y Fm(Fig.)e(17.)g Fs(\(top\))g(P)o(ersp)q(ectiv)o(e)h(conic)g (diagrams,)g(\(b)q(ottom\))f Fe(COP)f Fs(pro)r(jections:)i(\(left\))e Fk(\022)1164 1425 y Ft(1)1192 1421 y Fs(=)g(20)1271 1406 y Ff(\016)1289 1421 y Fk(;)6 b(\022)1324 1425 y Ft(2)1352 1421 y Fs(=)11 b(70)1431 1406 y Ff(\016)1449 1421 y Fs(;)g(\(righ)o (t\))i Fk(\022)1611 1425 y Ft(1)1639 1421 y Fs(=)d Fk(\022)1697 1425 y Ft(2)1725 1421 y Fs(=)h(45)1804 1406 y Ff(\016)1822 1421 y Fs(;)g(b)q(oth)i(ha)o(v)o(e)-116 1467 y(the)g(same)g Fk(\022)64 1471 y Fd(a)96 1467 y Fs(and)g(b)q(oth)h(div)o(erge)g(at)f Fk(\022)f Fs(=)e Fk(\022)528 1471 y Fd(a)556 1467 y Fg(\006)e Fs(90)632 1451 y Ff(\016)-116 2545 y Fm(Fig.)14 b(18.)f Fe(COD)f Fs(pro)r(jection:)i(\(left\))f Fk(\022)422 2549 y Ft(1)450 2545 y Fs(=)d(20)528 2530 y Ff(\016)547 2545 y Fk(;)c(\022)582 2549 y Ft(2)610 2545 y Fs(=)k(70)688 2530 y Ff(\016)707 2545 y Fs(;)i(\(righ)o(t\))i Fk(\022)871 2549 y Ft(1)899 2545 y Fs(=)c Fk(\022)957 2549 y Ft(2)985 2545 y Fs(=)h(45)1064 2530 y Ff(\016)1082 2545 y Fs(;)i(b)q(oth)g(ha)o (v)o(e)h(the)f(same)g Fk(\022)1465 2549 y Fd(a)1497 2545 y Fs(and)h(no)f(limits)p eop %%Page: 17 17 17 16 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(17)-116 -93 y Fw(This)13 b(pro)r(jection)h(is)f(illustrated)g(in)g (Fig.)f(19)h(with)g(t)o(w)o(o)g(di\013eren)o(t)-116 -43 y(parameterizations.)g(These)i(expressions)h(remain)d(useful)h(when) -116 7 y Fo(\022)-97 13 y Fu(1)-64 7 y Fw(=)g Fo(\022)1 13 y Fu(2)36 7 y Fw(and)h(require)h(no)g(other)g(form.)d(The)j(k)o(eyw) o(ord)f Fp(PROJP1)-116 57 y Fw(will)10 b(b)q(e)j(used)g(to)e(giv)o(e)h Fo(\022)259 63 y Fi(a)291 57 y Fw(in)g(degrees)i(and)e(the)g(k)o(eyw)o (ord)g Fp(PROJP2)-116 106 y Fw(will)k(b)q(e)j(used)g(to)f(giv)o(e)f Fo(\013)h Fw(in)f(degrees.)j(The)e(default)g(v)n(alue)f(for)-116 156 y Fp(PROJP2)11 b Fw(will)g(b)q(e)i(0)f(while)g Fp(PROJP1)f Fw(has)h(no)h(default;)e(it)h(is)h(recom-)-116 206 y(mended)g(that)h(b) q(oth)g(k)o(eyw)o(ords)g(alw)o(a)o(ys)f(b)q(e)i(giv)o(en.)-116 353 y(4.3.4.)c(COO)k(\(orthomorphic\))-116 443 y(The)k(t)o(w)o (o-standard)f(orthomorphic)f(pro)r(jection)i(is)g(c)o(haracter-)-116 493 y(ized)13 b(b)o(y)f(t)o(w)o(o)g(angles,)g Fn(\000)p Fw(90)310 478 y Fj(\016)340 493 y Fn(\024)g Fo(\022)403 499 y Fu(1)434 493 y Fn(\024)f Fo(\022)496 499 y Fu(2)527 493 y Fn(\024)h Fw(90)613 478 y Fj(\016)632 493 y Fw(.)g(The)g(pro)r (jection)-116 543 y(form)o(ul\032)f(are)-114 673 y Fo(R)-82 679 y Fi(\022)-43 673 y Fw(=)22 b Fo( )46 615 y Fl(\024)68 673 y Fw(tan)135 615 y Fl(\022)170 645 y Fw(90)212 630 y Fj(\016)240 645 y Fn(\000)10 b Fo(\022)p 170 664 132 2 v 226 702 a Fw(2)307 615 y Fl(\023\025)360 623 y Fi(C)849 673 y Fw(\(79\))-116 760 y Fo(A)-85 766 y Fi(\036)-43 760 y Fw(=)22 b Fo(C)s(\036)780 b Fw(\(80\))-106 859 y Fo(Y)-82 865 y Fu(0)-43 859 y Fw(=)22 b Fo( )46 800 y Fl(\024)68 859 y Fw(tan)135 800 y Fl(\022)170 830 y Fw(90)212 815 y Fj(\016)240 830 y Fn(\000)10 b Fo(\022)301 836 y Fi(a)p 170 849 151 2 v 235 887 a Fw(2)326 800 y Fl(\023\025)379 809 y Fi(C)849 859 y Fw(\(81\))-116 951 y(where)-92 1028 y Fo( )22 b Fn(\021)16 1000 y Fw(180)79 985 y Fj(\016)p 16 1019 82 2 v 44 1057 a Fo(\031)219 1000 y Fw(cos)8 b Fo(\022)301 1006 y Fu(1)p 107 1019 326 2 v 107 1071 a Fo(C)146 1037 y Fl(\002)164 1071 y Fw(tan)231 1037 y Fl(\000)255 1054 y Fu(90)289 1045 y Fh(\016)305 1054 y Fj(\000)p Fi(\022)347 1058 y Fc(1)p 255 1061 109 2 v 300 1085 a Fu(2)368 1037 y Fl(\001\003)404 1044 y Fi(C)849 1028 y Fw(\(82\))-43 1159 y(=)16 1131 y(180)79 1116 y Fj(\016)p 16 1149 82 2 v 44 1187 a Fo(\031)219 1131 y Fw(cos)g Fo(\022)301 1137 y Fu(2)p 107 1149 326 2 v 107 1201 a Fo(C)146 1167 y Fl(\002)164 1201 y Fw(tan)231 1167 y Fl(\000)255 1184 y Fu(90)289 1176 y Fh(\016)305 1184 y Fj(\000)p Fi(\022)347 1188 y Fc(2)p 255 1191 109 2 v 300 1215 a Fu(2)368 1167 y Fl(\001\003)404 1174 y Fi(C)849 1159 y Fw(\(83\))-96 1323 y Fo(C)23 b Fn(\021)69 1280 y Fw(ln)110 1234 y Fl(\020)140 1263 y Fu(cos)5 b Fi(\022)206 1267 y Fc(2)p 140 1270 83 2 v 140 1294 a Fu(cos)g Fi(\022)206 1298 y Fc(1)227 1234 y Fl(\021)p 16 1313 290 2 v 16 1381 a Fw(ln)57 1323 y Fl(\024)84 1351 y Fu(tan)132 1320 y Fl(\000)156 1335 y Fc(90)184 1329 y Fh(\016)201 1335 y(\000)p Fb(\022)237 1341 y Fc(2)p 156 1344 99 2 v 198 1361 a(2)259 1320 y Fl(\001)p 84 1372 195 2 v 84 1406 a Fu(tan)132 1375 y Fl(\000)156 1390 y Fc(90)184 1384 y Fh(\016)201 1390 y(\000)p Fb(\022)237 1396 y Fc(1)p 156 1399 99 2 v 198 1416 a(2)259 1375 y Fl(\001)283 1323 y(\025)12 b Fo(:)520 b Fw(\(84\))-116 1476 y(Using)13 b(Eqs.)g(45)g(and)h(46,)f(the)h(in)o(v)o(erse)h (equations)f(are)-88 1561 y Fo(\036)20 b Fw(=)16 1533 y Fo(A)47 1539 y Fi(\036)p 16 1552 54 2 v 26 1590 a Fo(C)849 1561 y Fw(\(85\))-84 1679 y Fo(\022)i Fw(=)g(90)53 1662 y Fj(\016)80 1679 y Fn(\000)10 b Fw(2)d(tan)209 1662 y Fj(\000)p Fu(1)261 1608 y Fl( )294 1621 y(\024)321 1651 y Fo(R)353 1657 y Fi(\022)p 321 1670 51 2 v 332 1708 a Fo( )376 1621 y Fl(\025)398 1629 y Fu(\(1)p Fi(=C)r Fu(\))492 1608 y Fl(!)539 1679 y Fo(:)298 b Fw(\(86\))-116 1820 y(The)20 b(expression)h(for)f Fo(C)j Fw(b)q(ecomes)d(un)o(wieldy)f (when)i Fo(\022)790 1826 y Fu(1)831 1820 y Fw(=)h Fo(\022)904 1826 y Fu(2)-116 1870 y Fw(and)14 b(ma)o(y)f(b)q(e)j(replaced)g(with)e Fo(C)i Fw(=)e(sin)7 b Fo(\022)543 1876 y Fu(1)562 1870 y Fw(.)14 b(Expressions)j(in)d(the)-116 1920 y(a)o(v)o(erage)k(and)g (di\013erence)i(angles)e(app)q(ear)h(less)g(than)f(useful)g(in)-116 1970 y(this)11 b(case.)i(This)e(pro)r(jection)h(is)g(illustrated)f(in)g (Fig.)h(20)f(with)g(t)o(w)o(o)-116 2019 y(di\013eren)o(t)16 b(parameterizations.)f(The)h(k)o(eyw)o(ord)g Fp(PROJP1)f Fw(will)f(b)q(e)-116 2069 y(used)k(to)f(giv)o(e)g Fo(\022)144 2075 y Fi(a)182 2069 y Fw(in)g(degrees)i(and)f(the)g(k)o(eyw)o(ord)f Fp(PROJP2)f Fw(will)-116 2119 y(b)q(e)c(used)h(to)f(giv)o(e)f Fo(\013)h Fw(in)g(degrees.)h(The)f(default)g(v)n(alue)g(for)f Fp(PROJP2)-116 2169 y Fw(will)h(b)q(e)j(0)f(while)f Fp(PROJP1)g Fw(has)h(no)g(default;)f(it)h(is)g(recommended)-116 2219 y(that)g(b)q(oth)g(k)o(eyw)o(ords)g(alw)o(a)o(ys)f(b)q(e)h(giv)o(en.) -116 2366 y Fr(4.4.)g(Conventional)-116 2456 y(Conventional)h Fw(pro)r(jections)g(are)g(mathematical)d(constructions)-116 2506 y(designed)e(to)g(map)e(the)i(en)o(tire)h(sphere)g(with)e(minimal) d(distortion.)-116 2556 y(By)18 b(con)o(v)o(en)o(tion,)f(the)i (reference)i(p)q(oin)o(t)d(in)f(all)g(of)h(the)h(con)o(v)o(en-)-116 2605 y(tional)14 b(pro)r(jections)j(is)e(the)i(nativ)o(e)e(co)q (ordinate)h(system)g(origin,)-116 2655 y(\()p Fo(\036;)7 b(\022)q Fw(\))k(=)h(\(0)p Fo(;)7 b Fw(0\).)970 -93 y(4.4.1.)12 b(BON)j(\(Bonne's)f(equal)g(area\))970 -16 y(Bonne's)19 b(equal-area)f(p)q(olyconic)f(pro)r(jection)i(is)f(c)o(haracterized)970 34 y(b)o(y)d(a)f(single)g(angle,)g Fn(\000)p Fw(90)1375 19 y Fj(\016)1407 34 y Fn(\024)f Fo(\022)1471 40 y Fu(1)1503 34 y Fn(\024)g Fw(90)1590 19 y Fj(\016)1608 34 y Fw(.)i(P)o(arallels)e (are)i(concen-)970 83 y(tric)g(equidistan)o(t)f(arcs)i(of)e(circles)h (of)f(true)i(length.)e(The)h(pro)r(jec-)970 133 y(tion)f(form)o(ul\032) d(are)1000 208 y Fo(x)20 b Fw(=)h Fo(R)1129 214 y Fi(\022)1154 208 y Fw(sin)7 b Fo(A)1243 214 y Fi(\036)1936 208 y Fw(\(87\))1002 270 y Fo(y)22 b Fw(=)f Fo(Y)1121 276 y Fu(0)1149 270 y Fn(\000)10 b Fo(R)1223 276 y Fi(\022)1248 270 y Fw(cos)d Fo(A)1341 276 y Fi(\036)1370 270 y Fo(:)554 b Fw(\(88\))970 332 y(where)981 410 y Fo(Y)1005 416 y Fu(0)1044 410 y Fw(=)1102 382 y(180)1165 367 y Fj(\016)p 1102 400 82 2 v 1130 438 a Fo(\031)1195 410 y Fw(cot)8 b Fo(\022)1277 416 y Fu(1)1305 410 y Fw(+)h Fo(\022)1365 416 y Fu(1)973 484 y Fo(R)1005 490 y Fi(\022)1044 484 y Fw(=)21 b Fo(Y)1121 490 y Fu(0)1149 484 y Fn(\000)10 b Fo(\022)970 563 y(A)1001 569 y Fi(\036)1044 563 y Fw(=)1108 535 y Fo(\036)d Fw(cos)h Fo(\022)p 1102 554 128 2 v 1130 575 a Fi(\031)p 1107 582 67 2 v 1107 606 a Fu(180)1158 598 y Fh(\016)1179 592 y Fo(R)1211 598 y Fi(\022)1241 563 y Fo(:)970 652 y Fw(The)15 b(in)o(v)o(erse)f(form)o(ul\032)e(are)i(then)973 718 y Fo(R)1005 724 y Fi(\022)1044 718 y Fw(=)21 b(sign)7 b(\()p Fo(\022)1211 724 y Fu(1)1230 718 y Fw(\))1253 680 y Fl(p)p 1295 680 260 2 v 38 x Fo(x)1319 706 y Fu(2)1346 718 y Fw(+)j(\()p Fo(Y)1428 724 y Fu(0)1456 718 y Fn(\000)f Fo(y)q Fw(\))1534 706 y Fu(2)970 805 y Fo(A)1001 811 y Fi(\036)1044 805 y Fw(=)21 b(arg)1169 747 y Fl(\022)1205 777 y Fo(Y)1229 783 y Fu(0)1257 777 y Fn(\000)9 b Fo(y)p 1205 796 116 2 v 1237 834 a(R)1269 840 y Fi(\022)1325 805 y Fo(;)1362 777 y(x)p 1349 796 51 2 v 1349 834 a(R)1381 840 y Fi(\022)1404 747 y Fl(\023)1003 893 y Fo(\022)22 b Fw(=)f Fo(Y)1121 899 y Fu(0)1149 893 y Fn(\000)10 b Fo(R)1223 899 y Fi(\022)1936 893 y Fw(\(89\))999 966 y Fo(\036)20 b Fw(=)1130 937 y Fo(\031)p 1102 956 82 2 v 1102 994 a Fw(180)1165 982 y Fj(\016)1188 966 y Fo(A)1219 972 y Fi(\036)1242 966 y Fo(R)1274 972 y Fi(\022)1299 966 y Fo(=)7 b Fw(cos\()p Fo(\022)q Fw(\))g Fo(:)483 b Fw(\(90\))970 1052 y(This)24 b(pro)r(jection)h(is)f(illustrated)g(in) g(Fig.)12 b(21.)23 b(The)i(k)o(eyw)o(ord)970 1101 y Fp(PROJP1)14 b Fw(will)g(b)q(e)i(used)g(to)f(giv)o(e)f Fo(\022)1507 1107 y Fu(1)1541 1101 y Fw(in)h(degrees)i(and)e(has)g(no)g(de-)970 1151 y(fault.)f(The)i(case)g Fo(\022)1276 1157 y Fu(1)1309 1151 y Fw(=)e(0)h(giv)o(es)g(a)g(div)o(ergen)o(t)g(equation)g(for)g Fo(R)1980 1157 y Fi(\022)1998 1151 y Fw(,)970 1201 y(but)f(not)g(for)g (\()p Fo(x;)7 b(y)q Fw(\).)13 b(This)h(case)h(should)f(b)q(e)g(written) h(as)f(the)g Fp(GLS)970 1251 y Fw(pro)r(jection)e(\(whic)o(h)f(it)f (is\))h(rather)h(than)f(a)g Fp(BON)g Fw(in)f(order)i(to)f(a)o(v)o(oid) 970 1301 y(p)q(osing)16 b(an)o(y)g(mathematical)d(di\016culties.)i(T)m (o)h(do)g(otherwise)h(is,)970 1351 y(w)o(e)d(b)q(eliev)o(e,)g(w)o(orse) h(than)e(simply)f(\\bad)i(form.")970 2572 y Fm(Fig.)g(21.)g Fe(BON)d Fs(pro)r(jection:)j Fk(\022)1411 2576 y Ft(1)1439 2572 y Fs(=)d(45)1518 2556 y Ff(\016)1536 2572 y Fs(,)i(no)g(limits)p eop %%Page: 18 18 18 17 bop -116 -192 a Fs(18)430 b(Greisen)14 b(&)e(Calabretta:)i (Represen)o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 703 y Fm(Fig.)g(19.)f Fe(COE)f Fs(pro)r(jection:)i(\(left\))f Fk(\022)422 707 y Ft(1)450 703 y Fs(=)d(20)528 687 y Ff(\016)547 703 y Fk(;)c(\022)582 707 y Ft(2)610 703 y Fs(=)k(70)688 687 y Ff(\016)707 703 y Fs(;)i(\(righ)o(t\))i Fk(\022)871 707 y Ft(1)899 703 y Fs(=)c Fk(\022)957 707 y Ft(2)985 703 y Fs(=)h(45)1064 687 y Ff(\016)1082 703 y Fs(;)i(b)q(oth)g(ha)o(v)o(e)h(the)f(same)g Fk(\022)1465 707 y Fd(a)1497 703 y Fs(and)h(no)f(limits)-116 1599 y Fm(Fig.)h(20.)f Fe(COO)f Fs(pro)r(jection:)i(\(left\))f Fk(\022)422 1603 y Ft(1)450 1599 y Fs(=)d(20)528 1583 y Ff(\016)547 1599 y Fk(;)c(\022)582 1603 y Ft(2)610 1599 y Fs(=)k(70)688 1583 y Ff(\016)707 1599 y Fs(;)i(\(righ)o(t\))i Fk(\022)871 1603 y Ft(1)899 1599 y Fs(=)c Fk(\022)957 1603 y Ft(2)985 1599 y Fs(=)h(45)1064 1583 y Ff(\016)1082 1599 y Fs(;)i(b)q(oth)g(ha)o(v)o(e)h(the)f(same)g Fk(\022)1465 1603 y Fd(a)1497 1599 y Fs(and)h(div)o(erge)g(at)f Fk(\022)f Fs(=)e Fg(\000)p Fs(90)1889 1583 y Ff(\016)-116 1733 y Fw(4.4.2.)h(PCO)k(\(p)q(olyconic\))-116 1809 y(In)21 b(the)h(p)q(olyconic)g(pro)r(jection,)f(ev)o(ery)h(parallel)f(is)g(pro) r(jected)-116 1859 y(as)g(arcs)h(of)e(circles)i(of)f(radius)g Fo(r)8 b Fw(cot)f Fo(\022)23 b Fw(at)e(their)g(true)h(length,)-116 1909 y(2)p Fo(\031)q(r)7 b Fw(cos)h Fo(\022)q Fw(,)15 b(and)f(correctly)i(divided.)e(The)h(scale)g(along)f(the)h(cen-)-116 1959 y(tral)j(meridian)f(is)i(true)h(and)f(consequen)o(tly)h(the)f (parallels)f(are)-116 2009 y(not)c(concen)o(tric.)h(The)g(pro)r (jection)g(form)o(ul\032)d(m)o(ust)i(then)h(b)q(e)g(ex-)-116 2059 y(pressed)g(in)f Fo(x)f Fw(and)h Fo(y)i Fw(and)e(are)-116 2161 y Fo(x)20 b Fw(=)-14 2133 y(180)49 2118 y Fj(\016)p -14 2152 82 2 v 14 2190 a Fo(\031)79 2161 y Fw(cot)8 b Fo(\022)g Fw(sin\()p Fo(\036)f Fw(sin)g Fo(\022)q Fw(\))487 b(\(91\))-115 2263 y Fo(y)23 b Fw(=)e Fo(\022)11 b Fw(+)57 2235 y(180)120 2220 y Fj(\016)p 57 2253 V 86 2291 a Fo(\031)151 2263 y Fw(cot)c Fo(\022)i Fw([)o(1)g Fn(\000)h Fw(cos\()p Fo(\036)d Fw(sin)g Fo(\022)q Fw(\)])14 b Fo(:)290 b Fw(\(92\))-116 2358 y(The)17 b(in)o(v)o(erse)g(ma)o(y)d(not)j(b)q(e)g(expressed)i(so)d (simply)m(,)e(since)j(it)f(re-)-116 2408 y(quires)e(the)g(solution)g (of)f(the)h(transcenden)o(tal)i(equation)-116 2510 y Fo(x)-92 2493 y Fu(2)-65 2510 y Fn(\000)-18 2482 y Fw(360)45 2467 y Fj(\016)p -18 2501 V 10 2539 a Fo(\031)68 2510 y Fw(\()p Fo(y)11 b Fn(\000)f Fo(\022)q Fw(\))d(cot)h Fo(\022)j Fw(+)e(\()p Fo(y)i Fn(\000)f Fo(\022)q Fw(\))459 2493 y Fu(2)490 2510 y Fw(=)h(0)295 b(\(93\))-116 2605 y(One)17 b(ma)o(y)d(determine)i(a)g(solution)g(iterativ)o(ely)g(with)g (Newton's)-116 2655 y(metho)q(d)h(\(see)k(Section)12 b(6\))19 b(or,)f(p)q(erhaps)i(more)e(simply)m(,)d(b)o(y)k(as-)970 1733 y(suming)1007 1819 y Fo(\022)1026 1825 y Fu(0)1066 1819 y Fw(=)i(1)1140 1802 y Fj(\016)1166 1819 y Fw(sign\()p Fo(y)q Fw(\))970 1881 y(and)14 b(computing)970 1983 y Fo(\022)989 1989 y Fi(i)p Fu(+1)1066 1983 y Fw(=)21 b(tan)1179 1966 y Fj(\000)p Fu(1)1231 1912 y Fl( )1291 1936 y Fu(360)1342 1924 y Fh(\016)p 1291 1943 67 2 v 1314 1967 a Fi(\031)1363 1952 y Fw(\()p Fo(y)11 b Fn(\000)f Fo(\022)1471 1958 y Fi(i)1485 1952 y Fw(\))p 1269 1973 250 2 v 1269 2011 a Fo(x)1293 1999 y Fu(2)1320 2011 y Fw(+)g(\()p Fo(y)h Fn(\000)e Fo(\022)1469 2017 y Fi(i)1484 2011 y Fw(\))1500 1999 y Fu(2)1523 1912 y Fl(!)970 2104 y Fw(for)14 b(iteration)f Fo(i)p Fw(.)h(Once)h Fo(\022)g Fw(is)f(kno)o(wn,)f(then)970 2213 y Fo(\036)e Fw(=)1084 2185 y(1)p 1055 2204 79 2 v 1055 2242 a(sin)c Fo(\022)1146 2213 y Fw(arg)1218 2155 y Fl(\022)1253 2185 y Fw(180)1316 2170 y Fj(\016)p 1253 2204 82 2 v 1281 2242 a Fo(\031)1349 2213 y Fn(\000)i Fw(\()p Fo(y)j Fn(\000)d Fo(\022)q Fw(\))e(tan)h Fo(\022)q(;)f(x)g Fw(tan)f Fo(\022)1746 2155 y Fl(\023)1791 2213 y Fo(:)970 2322 y Fw(A)14 b(metho)q(d)f(of)g(doing)g(this)h(iteration)g(with)f(b)q (etter)j(con)o(v)o(ergence)970 2372 y(prop)q(erties)c(is)e(detailed)g (in)g(App)q(endix)j(D.2.)c(The)i(p)q(olyconic)f(pro-)970 2422 y(jection)k(is)g(illustrated)g(in)f(Fig.)e(22.)970 2529 y(4.4.3.)h(GLS)h(\(Sanson-Flamsteed,)g(sin)o(usoidal\))970 2605 y(P)o(arallels)d(are)g(equispaced)h(and)f(their)h(length)f(is)g(c) o(hosen)h(to)f(mak)o(e)970 2655 y(an)j(equal)g(area)g(pro)r(jection.)g (It)h(is,)e(in)h(fact,)g(the)g(equatorial)g(case)p eop %%Page: 19 19 19 18 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(19)-116 761 y Fm(Fig.)14 b(22.)f Fe(PCO)f Fs(pro)r(jection,)i(no)f (limits)-116 893 y Fw(\()p Fo(\022)-81 899 y Fu(1)-51 893 y Fw(=)f(0\))i(of)f(Bonne's)i(pro)r(jection.)e(The)i(form)o(ul\032) c(are)-115 985 y Fo(x)20 b Fw(=)h Fo(\036)7 b Fw(cos)g Fo(\022)754 b Fw(\(94\))-114 1047 y Fo(y)23 b Fw(=)e Fo(\022)9 b(;)827 b Fw(\(95\))-116 1109 y(whic)o(h)13 b(rev)o(erse)j(in)o(to)-116 1176 y Fo(\036)k Fw(=)17 1148 y Fo(x)p -13 1166 85 2 v -13 1204 a Fw(cos)8 b Fo(y)849 1176 y Fw(\(96\))-112 1260 y Fo(\022)22 b Fw(=)f Fo(y)9 b(:)826 b Fw(\(97\))-116 1352 y(This)13 b(pro)r(jection)i(is)e (illustrated)h(in)f(Fig.)f(23.)-116 1461 y(4.4.4.)f(P)m(AR)j(\(parab)q (olic)f(pro)r(jection\))-116 1537 y(In)29 b(the)h(parab)q(olic)e(or)h (Craster)i(pro)r(jection,)e(parallels)f(are)-116 1587 y(straigh)o(t)15 b(lines)h(parallel)f(to)h(the)g(equator)h(while)e (meridians)f(are)-116 1637 y(parab)q(olic)19 b(arcs.)i(The)g(scaling)f (is)g(set)i(to)e(mak)o(e)f(this)h(also)g(an)-116 1687 y(equal-area)13 b(pro)r(jection.)h(The)g(form)o(ul\032)e(are)-115 1801 y Fo(x)20 b Fw(=)h Fo(\036)14 1743 y Fl(\022)44 1801 y Fw(2)7 b(cos)139 1773 y(2)p Fo(\022)p 139 1792 42 2 v 150 1830 a Fw(3)195 1801 y Fn(\000)j Fw(1)258 1743 y Fl(\023)849 1801 y Fw(\(98\))-114 1914 y Fo(y)23 b Fw(=)e(180)45 1897 y Fj(\016)70 1914 y Fw(sin)133 1886 y Fo(\022)p 133 1905 21 2 v 133 1943 a Fw(3)166 1914 y Fo(;)671 b Fw(\(99\))-116 1988 y(whic)o(h)13 b(has)h(in)o(v)o(erse)h (form)o(ul\032)-112 2055 y Fo(\022)22 b Fw(=)f(3)7 b(sin)61 2037 y Fj(\000)p Fu(1)112 2009 y Fl(\020)172 2027 y Fo(y)p 142 2045 82 2 v 142 2083 a Fw(180)205 2071 y Fj(\016)228 2009 y Fl(\021)828 2055 y Fw(\(100\))-116 2156 y Fo(\036)20 b Fw(=)-13 2128 y(180)50 2113 y Fj(\016)p -13 2147 V 15 2185 a Fo(\031)200 2128 y(x)p 78 2147 268 2 v 78 2185 a Fw(1)9 b Fn(\000)h Fw(4\()p Fo(y)q(=)p Fw(180)292 2173 y Fj(\016)311 2185 y Fw(\))327 2173 y Fu(2)357 2156 y Fo(:)459 b Fw(\(101\))-116 2270 y(This)13 b(pro)r(jection)i(is)e (illustrated)h(in)f(Fig.)f(24.)-116 2379 y(4.4.5.)f(AIT)j (\(Hammer-Aito\013)s(\))-116 2456 y(This)f(pro)r(jection)i(is)f(dev)o (elop)q(ed)g(from)f(the)h(zenithal)g(equal)g(area)-116 2506 y(pro)r(jection)f(b)o(y)g(rotating)f(the)i(co)q(ordinates)f(to)g (the)h(equator)f(and)-116 2556 y(then)g(doubling)e(the)j(equatorial)e (scale)h(and)f(longitude)g(co)o(v)o(erage)-116 2605 y(\(Calabretta,)j (1992)o(\).)g(The)h(whole)f(sphere)i(is)e(mapp)q(ed)g(thereb)o(y)-116 2655 y(while)c(preserving)i(the)g(equal)f(area)g(prop)q(ert)o(y)m(.)g (The)g(form)o(ul\032)e(for)970 -93 y(the)h(pro)r(jection)f(and)g(its)g (in)o(v)o(erse)h(are)f(deriv)o(ed)h(in)f(Greisen)g(\(1986)o(\))970 -43 y(and)k(Calabretta)g(\(1992)o(\))g(among)d(others.)k(They)f(are)978 71 y Fo(x)20 b Fw(=)h(2)p Fo(\013)7 b Fw(cos)g Fo(\022)i Fw(sin)1283 43 y Fo(\036)p 1283 61 25 2 v 1285 99 a Fw(2)1915 71 y(\(102\))980 147 y Fo(y)22 b Fw(=)f Fo(\013)7 b Fw(sin)g Fo(\022)729 b Fw(\(103\))970 210 y(where)975 306 y Fo(\013)20 b Fn(\021)1080 278 y Fw(180)1143 263 y Fj(\016)p 1080 297 82 2 v 1108 335 a Fo(\031)1167 234 y Fl(s)p 1208 234 326 2 v 1361 278 a Fw(2)p 1213 297 316 2 v 1213 335 a(1)9 b(+)h(cos)d Fo(\022)i Fw(cos\()p Fo(\036=)p Fw(2\))1541 306 y Fo(:)362 b Fw(\(104\))970 403 y(The)15 b(rev)o(erse)g(equations)f (are)977 492 y Fo(\036)20 b Fw(=)h(2)7 b(arg)1175 433 y Fl(\022)1206 492 y Fw(2)p Fo(Z)1258 475 y Fu(2)1286 492 y Fn(\000)i Fw(1)p Fo(;)1399 464 y(\031)p 1371 482 82 2 v 1371 520 a Fw(180)1434 508 y Fj(\016)1463 464 y Fo(Z)p 1463 482 32 2 v 1468 520 a Fw(2)1499 492 y Fo(x)1523 433 y Fl(\023)1915 492 y Fw(\(105\))981 594 y Fo(\022)22 b Fw(=)f(sin)1126 576 y Fj(\000)p Fu(1)1171 594 y Fw(\()1220 566 y Fo(\031)p 1192 584 82 2 v 1192 622 a Fw(180)1255 610 y Fj(\016)1278 594 y Fo(y)q(Z)s Fw(\))569 b(\(106\))970 668 y(where)970 758 y Fo(Z)24 b Fn(\021)1075 692 y Fl(r)p 1117 692 507 2 v 66 x Fw(1)9 b Fn(\000)1188 712 y Fl(\020)1246 730 y Fo(\031)p 1218 749 82 2 v 1218 787 a Fw(180)1281 775 y Fj(\016)1309 730 y Fo(x)p 1309 749 24 2 v 1310 787 a Fw(4)1338 712 y Fl(\021)1363 721 y Fu(2)1391 758 y Fn(\000)1432 712 y Fl(\020)1490 730 y Fo(\031)p 1462 749 82 2 v 1462 787 a Fw(180)1525 775 y Fj(\016)1553 730 y Fo(y)p 1553 749 22 2 v 1554 787 a Fw(2)1580 712 y Fl(\021)1605 721 y Fu(2)1630 758 y Fo(:)273 b Fw(\(107\))970 871 y(This)14 b(pro)r(jection)g(is)g(illustrated)g(in)f(Fig.)e(25.)1688 856 y Fu(3)970 981 y Fw(4.4.6.)h(MOL)i(\(Mollw)o(eide's\))970 1059 y(In)j(Mollw)o(eide's)e(pro)r(jection,)h(meridians)f(are)i(pro)r (jected)h(as)f(el-)970 1109 y(lipses,)i(parallels)f(as)h(straigh)o(t)g (lines)g(with)g(spacing)g(c)o(hosen)h(to)970 1158 y(mak)o(e)13 b(the)h(pro)r(jection)g(equal)g(area.)f(The)i(form)o(ul\032)c(are)1025 1279 y Fo(x)20 b Fw(=)1127 1251 y(2)1148 1216 y Fn(p)p 1183 1216 21 2 v 35 x Fw(2)p 1127 1269 77 2 v 1153 1307 a Fo(\031)1209 1279 y(\036)7 b Fw(cos)g Fo(\013)585 b Fw(\(108\))1027 1381 y Fo(y)22 b Fw(=)1123 1344 y Fn(p)p 1157 1344 21 2 v 1157 1381 a Fw(2)1183 1353 y(180)1246 1338 y Fj(\016)p 1183 1371 82 2 v 1211 1409 a Fo(\031)1276 1381 y Fw(sin)7 b Fo(\013)g(;)535 b Fw(\(109\))970 1455 y(whic)o(h)14 b(rev)o(erse)i(to)1024 1540 y Fo(\036)k Fw(=)i Fo(\031)q(x=)1200 1481 y Fl(\022)1230 1540 y Fw(2)1251 1479 y Fl(r)p 1292 1479 236 2 v 1292 1540 a Fw(2)9 b Fn(\000)g Fw(\()1413 1512 y Fo(\031)p 1384 1530 82 2 v 1384 1568 a Fw(180)1447 1556 y Fj(\016)1471 1540 y Fo(y)q Fw(\))1508 1528 y Fu(2)1534 1481 y Fl(\023)1915 1540 y 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1968 y(sin)7 b(2)p Fo(\013)p 1249 1987 106 2 v 1289 2025 a(\031)1366 1996 y(:)537 b Fw(\(112\))970 2100 y(This)14 b(pro)r(jection)g(is)g (illustrated)g(in)f(Fig.)e(26.)970 2211 y(4.4.7.)h(CSC)i(\(Cob)q(e)g (quadrilateralized)f(spherical)i(cub)q(e\))970 2288 y(The)21 b(quad-sphere)g(pro)r(jection)f(w)o(as)g(dev)o(elop)q(ed)g(to)g (represen)o(t)970 2338 y(the)14 b(full)f(sphere)i(in)e(6)g(separate)i (square)f(faces,)g(minim)o(i)o(zing)d(dis-)970 2388 y(tortions)h(while) f(main)o(taining)d(the)k(equal)f(area)h(prop)q(ert)o(y)m(.)f(A)h(new) 970 2438 y(axis)18 b(t)o(yp)q(e)g(is)g(required)h(to)f(describ)q(e)h (whic)o(h)f(of)g(the)g(6)g(faces)h(is)p 970 2474 237 2 v 970 2502 a Ft(3)1028 2518 y Fs(This)f(pro)r(jections)h(should)h(b)q (e)e(attributed)h(to)e(Hammer)h(rather)970 2564 y(than)12 b(Aito\013)g(according)g(to)f(Jones)h(\(1993\).)f(The)g(string)h Fe(AIT)d Fs(has)j(b)q(een)970 2610 y(in)i(use)e(to)q(o)h(long)h(within) 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1153 y Fi(j)1303 1147 y Fw(\))p Fo(;)g(f)t Fw(\()p Fo(y)1398 1153 y Fi(j)1417 1147 y Fo(;)g(x)1460 1153 y Fi(i)1473 1147 y Fw(\)\))i Fn(\000)h Fo(x)1580 1153 y Fi(i)1593 1147 y Fw(\))1609 1126 y Fu(2)1113 1219 y Fw(+)g(\()p Fo(F)c Fw(\()p Fo(f)t Fw(\()p Fo(y)1280 1225 y Fi(j)1298 1219 y Fo(;)h(x)1341 1225 y Fi(i)1354 1219 y Fw(\))p Fo(;)g(f)t Fw(\()p Fo(x)1453 1225 y Fi(i)1467 1219 y Fo(;)g(y)1506 1225 y Fi(j)1523 1219 y Fw(\)\))j Fn(\000)f Fo(y)1626 1225 y Fi(j)1644 1219 y Fw(\))1660 1198 y Fu(2)1693 1219 y Fo(:)970 1313 y Fw(The)16 b(Cob)q(e)g (parameterization)e(pro)q(duces)j(an)e(a)o(v)o(erage)g(error)h(of)970 1363 y(4.7)f(arc)h(seconds)h(o)o(v)o(er)e(the)h(full)f(\014eld.)g(The)h (ro)q(ot)f(mean)g(square)970 1413 y(and)h(p)q(eak)g(errors)h(are)f(6.6) f(and)h(24)f(arc)h(seconds,)h(resp)q(ectiv)o(ely)m(.)970 1463 y(In)k(the)g(cen)o(tral)g(parts)g(of)e(the)i(image)e(\()p Fn(j)p Fo(X)s Fn(j)p Fo(;)7 b Fn(j)p Fo(Y)h Fn(j)22 b(\024)h Fw(0)p Fo(:)p Fw(8\),)c(the)970 1513 y(a)o(v)o(erage)d(and)g(ro)q(ot)g (mean)f(square)i(errors)g(are)g(5.9)e(and)h(8.0)f(arc)970 1562 y(seconds,)g(larger)f(than)g(for)f(the)i(full)e(\014eld.)g(One)i (ma)o(y)d(also)h(write)970 1612 y(Eq.)f(118)e(in)f(the)i(form)d(of)i (Eq.)i(121)d(and)h(then)h(determine)f(optimal)970 1662 y(parameters)h(equiv)n(alen)o(t)f(to)g(the)h Fo(P)1518 1668 y Fi(ij)1558 1662 y Fw(to)f(minim)o(ize,)e(sa)o(y)m(,)h(the)j(sum) 970 1712 y(of)e Fo(E)1047 1697 y Fu(2)1045 1723 y Fi(ij)1085 1712 y Fw(o)o(v)o(er)g(the)h(full)f(\014eld.)g(The)h(Cob)q(e)g(form)o (ul\032)d(pro)q(duce)k(errors)970 1762 y(similar)h(to)i Fo(N)k Fw(=)14 b(4.)g(Limiting)e(ourselv)o(es)17 b(to)e Fo(N)j Fw(=)c(6,)h(as)g(in)g(the)970 1811 y(Cob)q(e)i(usage)f(of)f(Eq.) d(121,)j(c)o(hanges)i(the)f(error)h(to)f(a)o(v)o(erage)g(3.1,)970 1861 y(ro)q(ot)10 b(mean)f(square)i(5.4,)d(and)i(p)q(eak)g(22)f(arc)i (seconds)g(o)o(v)o(er)f(the)h(full)970 1911 y(\014eld.)f(Allo)o(wing)e Fo(N)16 b Fw(=)c(9,)d(reduces)j(these)g(n)o(um)o(b)q(ers)d(to)h(0.74,)e (0.92,)970 1961 y(and)17 b(9.6)f(arc)i(seconds,)g(resp)q(ectiv)o(ely)m (.)g(In)f(the)h(cen)o(tral)g(regions,)970 2011 y(the)f(p)q(eak)g (errors)g(drop)g(to)f(14)g(and)g(2)g(arc)h(seconds)g(for)f Fo(N)21 b Fw(=)16 b(6)970 2060 y(and)e(9.)970 2177 y(4.4.8.)e(QSC)i (\(quadrilateralized)g(spherical)g(cub)q(e\))970 2257 y(O'Neill)20 b(and)h(Laubsc)o(her)h(\(1976)o(\))f(deriv)o(ed)g(an)g (exact)g(expres-)970 2307 y(sion)f(for)g(an)g(equal-area)g (transformation)e(from)h(a)h(sphere)i(to)970 2356 y(the)d(six)g(faces)g (of)f(a)g(cub)q(e.)h(A)o(t)g(that)g(time,)e(their)i(form)o(ulatio)o(n) 970 2406 y(w)o(as)10 b(though)o(t)f(to)h(b)q(e)g(computationally)c(in)o (tractable,)j(but,)h(to)q(da)o(y)m(,)970 2456 y(with)15 b(mo)q(dern)f(computers)i(and)e(telescop)q(es)k(of)c(higher)h(angular) 970 2506 y(resolution)c(than)g(Cob)q(e,)f(their)h(form)o(ulation)d(has) j(come)f(in)o(to)g(use.)970 2556 y(F)m(red)k(P)o(att)f(\(1993,)f(priv)n (ate)h(comm)o(unication\))d(has)k(pro)o(vided)f(us)970 2605 y(with)e(the)h(in)o(v)o(erse)g(of)f(the)h(O'Neill)e(and)h(Laubsc)o (her)i(form)o(ula)c(and)970 2655 y(their)14 b(expression)i(in)d (cartesian)i(co)q(ordinates.)p eop %%Page: 23 23 23 22 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(23)-116 1151 y Fm(Fig.)14 b(27.)f Fe(CSC)f Fs(pro)r(jection,)i(no)f (limits)-54 1282 y Fw(As)32 b(with)f(the)h Fp(CSC)f Fw(pro)r(jection,)g (the)h(new)g(axis)f(t)o(yp)q(e)-116 1332 y Fp('CUBEFACE')8 b Fw(as)i(the)h(v)n(alue)e(for)h Fp(CTYPE)p Fr(n)f Fw(is)h(required)h (to)f(describ)q(e)-116 1382 y(whic)o(h)j(of)h(the)h(6)e(faces)i(is)f(b) q(eing)g(view)o(ed)g(in)g(the)g(data)g(cub)q(e.)h(By)-116 1432 y(con)o(v)o(en)o(tion,)i(the)i(co)q(ordinate)f(v)n(alues)g(at)g (the)h(reference)h(p)q(oin)o(t)-116 1482 y(on)e(the)h(longitude)f(and)g (latitude)g(axes)h(are)g(for)f(the)h(origin)e(of)-116 1532 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726 1930 a Fw(3\()p Fo(N)g Fn(\000)12 b Fw(1\)\),)-116 1980 y(where)j Fo(N)j Fw(is)c(the)h(n)o(um) o(b)q(er)e(of)g(pixels)h(on)f(the)i(axis.)-54 2057 y(O'Neill)c(and)i (Laubsc)o(her)h(\(1976)o(\))e(suggest)i(a)e(useful)h(metho)q(d)-116 2107 y(to)23 b(compute)g(the)h(pro)r(jection.)f(F)m(rom)f(the)i(nativ)o (e)f(longitude)-116 2157 y(and)c(latitude,)g(\()p Fo(\036;)7 b(\022)q Fw(\),)19 b(one)h(computes)g(the)g(direction)g(cosines)-116 2207 y(b)o(y)g(Eq.)12 b(113.)19 b(Then,)h(if)g Fo(\032)g Fw(is)h(the)g(largest)f(of)g Fo(n;)7 b(l)q(;)g(m;)g Fn(\000)p Fo(l)q(;)g Fn(\000)p Fo(m;)-116 2257 y Fw(and)21 b Fn(\000)p Fo(n)p Fw(,)g(the)h(face)g(n)o(um)o(b)q(er)f(is)h(0)f(through)h(5,)e (resp)q(ectiv)o(ely)m(,)-116 2307 y(and)k(the)h(parametric)f(v)n (ariables)f Fo(\013)h Fw(and)h Fo(\014)i Fw(ma)o(y)22 b(b)q(e)j(found)-116 2356 y(from)12 b(T)m(able)g(1.)i(The)h(p)q (osition)e(v)n(ariables)h(within)g(eac)o(h)h(face)f(are)-116 2406 y(\()p Fo(x)-76 2412 y Fi(f)-55 2406 y Fo(;)7 b(y)-16 2412 y Fi(f)5 2406 y Fw(\),)13 b(with)g(\(0)p Fo(;)7 b Fw(0\))12 b(at)h(the)h(cen)o(ter,)h Fo(x)525 2412 y Fi(f)559 2406 y Fw(increases)g(to)e(the)h(left,)-116 2456 y(and)20 b Fo(y)-9 2462 y Fi(f)33 2456 y Fw(increases)i(up)o(w)o (ard,)e(in)g(the)h(usual)f(manner.)f(O'Neill)-116 2506 y(and)e(Laubsc)o(her's)h(deriv)n(ation)f(applies)g(only)f(in)h(the)h (quadran)o(t)-116 2556 y Fn(\000)p Fw(45)-42 2541 y Fj(\016)-2 2556 y Fn(\024)j Fo(\036)g Fn(\024)h Fw(45)193 2541 y Fj(\016)231 2556 y Fw(and)e(m)o(ust)f(b)q(e)h(re\015ected)i(in)o(to)d (the)h(other)-116 2605 y(quadran)o(ts.)d(This)h(has)g(the)h(a\013ect)g (of)e(making)f(the)i(pro)r(jection)-116 2655 y(non-di\013eren)o(table)g (along)g(the)h(diagonals.)d(P)o(att's)i(form)o(ul\032)e(in)970 1282 y(our)e(nomenclature)g(are)974 1389 y Fo(u)21 b Fw(=)g(45)p Fo(S)1148 1321 y Fl(s)p 1190 1321 276 2 v 1281 1361 a Fw(1)9 b Fn(\000)h Fo(\032)p 1195 1380 266 2 v 1195 1424 a Fw(1)f Fn(\000)g Fw(1)p Fo(=)1308 1388 y Fl(p)p 1349 1388 112 2 v 1349 1424 a Fw(2)g(+)h Fo(\020)1442 1412 y Fu(2)1915 1389 y Fw(\(122\))976 1533 y Fo(v)23 b Fw(=)1086 1505 y Fo(u)p 1077 1523 42 2 v 1077 1561 a Fw(15)1130 1462 y Fl(")1154 1533 y Fw(tan)1214 1516 y Fj(\000)p Fu(1)1266 1533 y Fw(\()p Fo(\020)s Fw(\))10 b Fn(\000)f Fw(sin)1421 1515 y Fj(\000)p Fu(1)1473 1462 y Fl( )1603 1505 y Fo(\020)p 1510 1523 206 2 v 1510 1532 a Fl(p)p 1552 1532 165 2 v 35 x Fw(2\(1)g(+)g Fo(\020)1681 1555 y Fu(2)1700 1567 y Fw(\))1721 1462 y Fl(!#)1792 1533 y Fo(;)111 b Fw(\(123\))970 1630 y(where)974 1711 y Fo(u)21 b Fn(\021)1072 1653 y Fl(\032)1103 1686 y Fo(x)f Fw(if)13 b Fn(j)p Fo(\013)p Fn(j)e Fo(>)h Fn(j)p Fo(\014)r Fn(j)1103 1736 y Fo(y)24 b Fw(if)13 b Fn(j)p Fo(\013)p Fn(j)e(\024)h(j)p Fo(\014)r Fn(j)976 1827 y Fo(v)23 b Fn(\021)1072 1769 y Fl(\032)1103 1802 y Fo(y)h Fw(if)13 b Fn(j)p Fo(\013)p Fn(j)e Fo(>)h Fn(j)p Fo(\014)r Fn(j)1103 1852 y Fo(x)20 b Fw(if)13 b Fn(j)p Fo(\013)p Fn(j)e(\024)h(j)p Fo(\014)r Fn(j)977 1944 y Fo(\020)24 b Fn(\021)1072 1885 y Fl(\032)1103 1918 y Fo(\014)r(=\013)d Fw(if)13 b Fn(j)p Fo(\013)p Fn(j)d Fo(>)i Fn(j)p Fo(\014)r Fn(j)1103 1968 y Fo(\013=\014)23 b Fw(if)13 b Fn(j)p Fo(\013)p Fn(j)d(\024)i(j)p Fo(\014)r Fn(j)970 2060 y Fo(S)24 b Fn(\021)1072 2001 y Fl(\032)1103 2035 y Fw(+1)d(if)13 b Fo(\013)e(>)h Fn(j)p Fo(\014)r Fn(j)h Fw(or)h Fo(\014)g(>)e Fn(j)p Fo(\013)p Fn(j)1103 2084 y(\000)p Fw(1)21 b(if)13 b Fn(\000)p Fo(\013)e(>)h Fn(j)p Fo(\014)r Fn(j)i Fw(or)g Fn(\000)p Fo(\014)g(>)e Fn(j)p Fo(\013)p Fn(j)970 2163 y Fw(The)j(in)o(v)o(erse)f(form)o (ul\032)e(are)i(then)970 2271 y Fo(\032)21 b Fw(=)g(1)9 b Fn(\000)1137 2225 y Fl(\020)1176 2243 y Fo(u)p 1167 2261 42 2 v 1167 2299 a Fw(45)1213 2225 y Fl(\021)1238 2233 y Fu(2)1264 2200 y Fl( )1296 2271 y Fw(1)g Fn(\000)1439 2243 y Fw(1)p 1373 2261 153 2 v 1373 2270 a Fl(p)p 1414 2270 112 2 v 1414 2306 a Fw(2)g(+)h Fo(\020)1507 2294 y Fu(2)1531 2200 y Fl(!)1915 2271 y Fw(\(124\))970 2398 y Fo(\020)24 b Fw(=)1147 2370 y(sin)o(\(15)p Fo(v)q(=u)p Fw(\))p 1070 2389 344 2 v 1070 2432 a(cos)q(\(15)p Fo(v)q(=u)p Fw(\))9 b Fn(\000)h Fw(1)p Fo(=)1359 2397 y Fn(p)p 1393 2397 21 2 v 1393 2432 a Fw(2)1915 2398 y(\(125\))970 2506 y(The)15 b(pro)r(jection)g(is)f(illustrated)h(in)f(Fig.)d(28)j (for)g(the)i(full)d(sphere)970 2556 y(in)h(the)g(natural)g(co)q (ordinates.)1032 2605 y(Although)c(this)f(pro)r(jection)i(pro)q(duces)g (pleasing)e(full-sky)g(im-)970 2655 y(ages,)14 b(it)g(is)h(clearly)f (di\016cult)g(to)g(use)h(in)f(practice.)h(The)g Fp(QSC)e Fw(has)p eop %%Page: 24 24 24 23 bop -116 -192 a Fs(24)430 b(Greisen)14 b(&)e(Calabretta:)i (Represen)o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 1151 y Fm(Fig.)g(28.)f Fe(QSC)f Fs(pro)r(jection,)i(no)f(limits)-116 1282 y Fw(most)f(of)g(the)i(pleasing)e(asp)q(ects)j(of)e(the)h Fp(CSC)e Fw(pro)r(jection,)h(plus)g(it)-116 1332 y(has)j(an)f(exact)i (in)o(v)o(erse)g(and)f(is)f(exactly)i(equal)e(area.)h(Ho)o(w)o(ev)o (er,)-116 1382 y(it)11 b(has)h(discon)o(tin)o(uous)g(deriv)n(ativ)o(e)f (along)f(the)j(diagonals)d(of)h(eac)o(h)-116 1432 y(face.)-116 1567 y(4.4.9.)g(TSC)j(\(tangen)o(tial)f(spherical)h(cub)q(e\))-116 1654 y(In)h(a)f(manner)g(similar)f(to)i(the)g(quad-sphere)h(pro)r (jections)g(\(Sec-)-116 1704 y(tions)c(4.4.7)j(and)j(4.4.8\),)d(one)i (ma)o(y)f(represen)o(t)j(the)f(full)e(sphere)-116 1753 y(in)e(6)i(separate)g(square)g(faces)g(with)f(a)g(gnomonic)f(pro)r (jection)h(in)-116 1803 y(eac)o(h)20 b(face.)g(The)g(new)g(axis)g(t)o (yp)q(e)g Fp('CUBEFACE')e Fw(as)i(the)h(v)n(alue)-116 1853 y(for)14 b Fp(CTYPE)p Fr(n)h Fw(is)g(required)h(to)f(describ)q(e)h (whic)o(h)f(of)g(the)g(6)g(faces)h(is)-116 1903 y(b)q(eing)g(view)o(ed) g(in)g(the)h(data)e(cub)q(e.)i(By)g(con)o(v)o(en)o(tion,)e(the)i(co)q (or-)-116 1953 y(dinate)d(v)n(alues)g(at)h(the)g(reference)i(p)q(oin)o (t)d(on)g(the)h(longitude)f(and)-116 2002 y(latitude)c(axes)i(are)g (for)f(the)h(origin)e(of)h(the)h(nativ)o(e)e(co)q(ordinates)i(of)-116 2052 y(the)g(pro)r(jection)f(whic)o(h)h(is)f(lo)q(cated)g(in)g(the)h (cen)o(tral)g(face)g(\(1\).)e(The)-116 2102 y(faces)i(along)f(the)i (nativ)o(e)e(equator)i(are)f(n)o(um)o(b)q(ered)g(4,)f(3,)g(2,)g(and)h (1)-116 2152 y(from)d(left)h(to)h(righ)o(t,)f(while)g(the)i (northernmost)e(is)h(face)g(n)o(um)o(b)q(er)f(0)-116 2202 y(and)k(the)h(southernmost)f(is)g(n)o(um)o(b)q(er)f(5)h(\(Jac)o(k) h(Saba,)e(1993,)g(pri-)-116 2252 y(v)n(ate)e(comm)o(unication\).)d(If)k (the)g(full)f(glob)q(e)g(is)h(represen)o(ted,)i(then)-116 2301 y(the)f(axis)f(dimensions)f(alone)h(determine)g(the)h(pixel)f(co)q (ordinates)-116 2351 y(of)i(the)h(reference)i(p)q(oin)o(t)e(\(in)f(the) i(n)o(umeric)e(cen)o(ter)i(of)e(eac)o(h)i(face)-116 2401 y(=)i(\()p Fo(N)f Fw(+)c(1)p Fo(:)p Fw(0\))p Fo(=)p Fw(2)p Fo(:)p Fw(0\))j(and)i(the)g(axis)g(incremen)o(ts)g(are,)g(within)f(a) -116 2451 y(sign,)10 b(360)41 2436 y Fj(\016)60 2451 y Fo(=)p Fw(\()p Fo(\031)122 2416 y Fn(p)p 156 2416 21 2 v 156 2451 a Fw(3\()p Fo(N)g Fn(\000)5 b Fw(1\)\),)12 b(where)g Fo(N)17 b Fw(is)12 b(the)g(n)o(um)o(b)q(er)f(of)g(pixels)-116 2501 y(on)i(the)i(axis.)-54 2556 y(O'Neill)c(and)i(Laubsc)o(her)h (\(1976)o(\))e(suggest)i(a)e(useful)h(metho)q(d)-116 2605 y(to)f(compute)f(the)i(pro)r(jection.)f(F)m(rom)e(the)j(nativ)o(e) f(longitude)f(and)-116 2655 y(latitude,)22 b(\()p Fo(\036;)7 b(\022)q Fw(\),)22 b(one)i(computes)e(the)i(direction)f(cosines)h(b)o (y)970 1282 y(Eq.)12 b(113.)17 b(Then,)h(if)f Fo(\032)h Fw(is)f(the)i(largest)f(of)f Fo(n;)7 b(l)q(;)g(m;)g Fn(\000)p Fo(l)q(;)g Fn(\000)p Fo(m;)17 b Fw(and)970 1332 y Fn(\000)p Fo(n)p Fw(,)d(the)g(face)g(n)o(um)o(b)q(er)f(is)g(0)h(through)f(5,)g (resp)q(ectiv)o(ely)m(,)h(and)g(the)970 1382 y(parametric)f(v)n (ariables)h Fo(\030)h Fw(and)f Fo(\021)h Fw(ma)o(y)d(b)q(e)j(found)e (from)f(T)m(able)g(1.)970 1432 y(The)j(p)q(osition)f(v)n(ariables)f (within)h(eac)o(h)h(face)f(are)h(\()p Fo(x)1805 1438 y Fi(f)1827 1432 y Fo(;)7 b(y)1866 1438 y Fi(f)1887 1432 y Fw(\),)14 b(with)970 1482 y(\(0)p Fo(;)7 b Fw(0\))18 b(at)g(the)i(cen)o(ter,)f Fo(x)1376 1488 y Fi(f)1416 1482 y Fw(increases)i(to)d(the)h(left,)f(and)h Fo(y)1921 1488 y Fi(f)1961 1482 y Fw(in-)970 1532 y(creases)d(up)o(w)o(ard,)d(in) h(the)g(usual)g(manner.)e(They)j(are)f(giv)o(en)f(b)o(y)997 1629 y Fo(x)1021 1635 y Fi(f)1063 1629 y Fw(=)21 b Fo(R)1148 1635 y Fi(\022)1164 1639 y Fb(f)1191 1629 y Fw(sin)7 b Fo(A)1280 1635 y Fi(\036)1300 1639 y Fb(f)1915 1629 y Fw(\(126\))1000 1692 y Fo(y)1020 1698 y Fi(f)1063 1692 y Fw(=)21 b Fn(\000)p Fo(R)1180 1698 y Fi(\022)1196 1702 y Fb(f)1223 1692 y Fw(cos)8 b Fo(A)1317 1698 y Fi(\036)1337 1702 y Fb(f)1365 1692 y Fo(;)538 b Fw(\(127\))970 1757 y(where)974 1823 y Fo(R)1006 1829 y Fi(\022)1022 1833 y Fb(f)1063 1823 y Fw(=)21 b(45)13 b(cot)1233 1790 y Fl(\000)1253 1823 y Fw(sin)1303 1806 y Fj(\000)p Fu(1)1348 1823 y Fw(\()p Fo(\032)p Fw(\))1401 1790 y Fl(\001)1915 1823 y Fw(\(128\))970 1886 y Fo(A)1001 1892 y Fi(\036)1021 1896 y Fb(f)1063 1886 y Fw(=)21 b(arg\()p Fn(\000)p Fo(\021)q(;)7 b(\030)r Fw(\))g Fo(:)597 b Fw(\(129\))970 1951 y(A)14 b(less)h(in)o(tuitiv)o(e,)d(but)i(simpler,)e(form)h(of)g(these)i(is)997 2037 y Fo(x)1021 2043 y Fi(f)1063 2037 y Fw(=)21 b(45)1170 2009 y Fo(\030)p 1169 2027 22 2 v 1169 2065 a(\032)1915 2037 y Fw(\(130\))1000 2136 y Fo(y)1020 2142 y Fi(f)1063 2136 y Fw(=)g(45)1169 2108 y Fo(\021)p 1169 2126 23 2 v 1169 2165 a(\032)1203 2136 y(:)700 b Fw(\(131\))970 2253 y(The)15 b(pro)r(jection)g(is)f(illustrated)h(in)f(Fig.)d(29)j (for)g(the)i(full)d(sphere)970 2303 y(in)g(the)h(natural)f(co)q (ordinates.)g(T)m(o)g(do)g(the)h(in)o(v)o(erse,)f(w)o(e)h(tak)o(e)f (the)970 2353 y(face)f(n)o(um)o(b)q(er,)f Fo(R)1245 2359 y Fi(\022)1261 2363 y Fb(f)1281 2353 y Fw(,)h(and)f Fo(A)1414 2359 y Fi(\036)1434 2363 y Fb(f)1455 2353 y Fw(,)h(and,)f(with)h(the)g (help)g(of)f(T)m(able)h(1,)970 2403 y(w)o(e)18 b(get)h(one)f(of)g(the)g (direction)g(cosines)h(and)f(the)h(ratio)e(of)h(the)970 2453 y(other)13 b(t)o(w)o(o)e(ma)o(y)m(,)f(thereb)o(y)m(,)i(determine)g (the)g(nativ)o(e)g(latitude)g(and)970 2503 y(longitude.)1032 2556 y(Although)e(this)f(pro)r(jection)i(pro)q(duces)g(pleasing)e (full-sky)g(im-)970 2605 y(ages,)14 b(it)g(is)h(clearly)f(di\016cult)g (to)g(use)h(in)f(practice.)h(The)g Fp(TSC)e Fw(has)970 2655 y(some)f(of)h(the)g(pleasing)f(asp)q(ects)j(of)d(the)i Fp(CSC)e Fw(pro)r(jection,)h(plus)g(it)p eop %%Page: 25 25 25 24 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(25)-116 1151 y Fm(Fig.)14 b(29.)f Fe(TSC)f Fs(pro)r(jection,)i(no)f (limits)-116 1282 y Fw(has)h(an)g(exact)h(in)o(v)o(erse)f(and)g(pro)r (jects)i(great)f(circles)g(in)o(to)e(piece-)-116 1332 y(wise)20 b(straigh)o(t)f(lines.)g(Ho)o(w)o(ev)o(er,)h(it)g(do)q(es)g (not)g(conserv)o(e)h(area,)-116 1382 y(length,)13 b(or)h(shap)q(e.)-116 1494 y Fx(5.)i(Con)o(v)o(erting)d(previous)g(formats)-116 1570 y Fw(There)18 b(exists)f(a)g(large)f(n)o(um)o(b)q(er)g(of)g(FITS)h (images)e(written)j(us-)-116 1620 y(ing)e(the)i(old)e(AIPS)i(con)o(v)o (en)o(tions)f(and)g(a)g(substan)o(tial)f(b)q(o)q(dy)h(of)-116 1670 y(soft)o(w)o(are)f(whic)o(h)g(understands)h(those)g(con)o(v)o(en)o (tions.)e(Although)-116 1720 y(the)d(NRA)o(O)g(and)g(other)h (institutions)f(will)f(probably)g(alter)h(their)-116 1770 y(soft)o(w)o(are)e(systems)h(to)g(understand)h(and)e(generate)i (the)g(new)f(con-)-116 1819 y(v)o(en)o(tions)16 b(prop)q(osed)h(here,)f (man)o(y)e(of)i(their)g(user)i(sites)e(will)f(not)-116 1869 y(b)q(e)10 b(able)g(to)g(up)q(date)h(their)g(v)o(ersions)g(of)e (AIPS,)h Fr(et)i(al.)g Fw(in)d(so)i(timely)-116 1919 y(a)k(manner.)f(Th)o(us,)i(for)f(a)h(few)f(y)o(ears,)h(FITS-writing)f (programs)-116 1969 y(whic)o(h)10 b(use)h(the)f(new)h(con)o(v)o(en)o (tions)f(should)g(also)g(write)g(the)h Fp(CROTA)-116 2019 y Fw(k)o(eyw)o(ord)h(whenev)o(er)j(p)q(ossible.)d(And,)h(of)f (course,)i(FITS-reading)-116 2068 y(programs)c(will)h(need)i(to)f (understand)h(the)g(old)e(con)o(v)o(en)o(tions)h(vir-)-116 2118 y(tually)19 b(forev)o(er.)h(If)g Fr(no)g Fp(PC)p Fr(nnnmmm)h Fw(k)o(eyw)o(ords)f(are)h(giv)o(en)e(in)-116 2168 y(the)c(header,)g(then,)g(for)f(longitude)g(axis)g Fo(i)h Fw(and)g(latitude)f(axis)g Fo(j)r Fw(,)-116 2218 y(reading)f(programs)g(should)h(set)-116 2269 y Fl(\022)-86 2302 y Fp(PC)p Fr(iiiiii)19 b Fp(PC)p Fr(iiijjj)-86 2352 y Fp(PC)p Fr(jjjiii)g Fp(PC)p Fr(jjjjjj)185 2269 y Fl(\023)227 2327 y Fw(=)-75 2385 y Fl(\022)-15 2418 y Fw(cos\()p Fp(CROTA)p Fr(j)5 b Fw(\))50 b Fn(\000)7 b Fw(sin\()p Fp(CROTA)p Fr(j)e Fw(\))k Fo(S)-44 2468 y Fw(sin\()p Fp(CROTA)p Fr(j)t Fw(\))e Fo(=)g(S)60 b Fw(cos\()p Fp(CROTA)p Fr(j)5 b Fw(\))547 2385 y Fl(\023)828 2444 y Fw(\(132\))-116 2556 y(where)23 b Fo(S)29 b Fw(=)d Fp(CDELT)p Fr(j)5 b Fo(=)i Fp(CDELT)p Fr(i)o Fw(.)21 b(W)m(e)h(note)h(that)g(the)g (expres-)-116 2605 y(sions)d(in)h(Hanisc)o(h)g(and)g(W)m(ells)e (\(1988\))h(and)h(Geldzahler)g(and)-116 2655 y(Sc)o(hlesinger)15 b(\(1992)o(\))f(are)h(for)e(the)i(pro)q(duct)g(of)f(the)g Fp(PC)p Fr(nnnmmm)970 1282 y Fw(matrix)e(and)h(the)h Fp(CDELT)p Fr(n)f Fw(matrices.)g(They)g(yield)g(the)h(same)f(re-)970 1332 y(sults)h(as)g(Eq.)e(132)h(for)g(the)i(usual)e(left-handed)h(sky)g (co)q(ordinates)970 1382 y(and)e(righ)o(t-handed)g(pixel)g(co)q (ordinates,)g(but)g(can)g(lead)g(to)g(an)g(in-)970 1432 y(correct)i(in)o(terpretation)e(\(namely)m(,)e(p)q(ossible)i(sign)g (errors)h(for)f(the)970 1482 y(cross)17 b(terms)f(in)g(the)h Fp(PC)e Fw(matrix\))g(for)g(other)i(con\014gurations)f(of)970 1532 y(the)e(co)q(ordinate)f(systems.)g(The)h(Hanisc)o(h)f(and)g(W)m (ells)g(draft)g(and)970 1581 y(the)h(co)q(ordinate)g(p)q(ortions)f(of)g (Geldzahler)g(and)g(Sc)o(hlesinger)h(are)970 1631 y(sup)q(erceded)j(b)o (y)d(the)h(presen)o(t)g(man)o(uscript.)e(In)h(the)h(presence)h(of)970 1681 y(rotation)g(or)g(sk)o(ew,)h(mo)q(dern)e(FITS-writing)h(programs)f (should)970 1731 y(alw)o(a)o(ys)k(write)i(the)g Fp(PC)p Fr(nnnmmm)g Fw(k)o(eyw)o(ords.)f(T)m(o)g(assist)g(older)970 1781 y(FITS-reading)i(programs,)d(for)j(a)f(few)h(y)o(ears,)f(these)i (mo)q(dern)970 1830 y(FITS-writing)15 b(programs)f(should)h(also)g (write)h Fp(CROTA)p Fr(j)e Fw(for)h(lati-)970 1880 y(tude)h(axis)e Fr(j)h Fw(when)g(the)g(longitude/latitude)f(pair)g(\014ts)i(Eq.)c(132.) 970 1930 y(The)i(\014t)g(to)f(Eq.)f(132)g(ma)o(y)g(b)q(e)i(determined)f (b)o(y)g(ev)n(aluating)f(b)q(oth)974 2021 y Fo(\032)995 2027 y Fi(i)1030 2021 y Fw(=)21 b(arg)7 b(\()p Fp(PC)p Fr(iiiiii)n Fo(;)g Fp(PC)p Fr(jjjiii)g Fo(S)r Fw(\))970 2083 y Fo(\032)991 2089 y Fi(j)1030 2083 y Fw(=)21 b(arg)7 b(\()p Fp(PC)p Fr(jjjjjj)n Fo(;)g Fn(\000)p Fp(PC)p Fr(iiijjj)e Fo(=)i(S)r Fw(\))14 b Fo(:)970 2174 y Fw(If)20 b Fo(\032)1039 2180 y Fi(i)1075 2174 y Fw(=)i Fo(\032)1150 2180 y Fi(j)1188 2174 y Fw(within)d(reasonable)i(precision)f(\(Geldzahler)h(and)970 2224 y(Sc)o(hlesinger,)12 b(1992)o(\),)f(then)h(the)g(older)f(k)o(eyw)o (ord)h(should)f(b)q(e)h(writ-)970 2274 y(ten)j(using)970 2365 y Fp(CROTA)p Fr(j)10 b Fw(=)i(\()p Fo(\032)1184 2371 y Fi(i)1208 2365 y Fw(+)d Fo(\032)1270 2371 y Fi(j)1288 2365 y Fw(\))p Fo(=)p Fw(2)569 b(\(133\))970 2456 y(where)15 b Fo(i)f Fw(is)g(the)h(longitude)e(axis)g(and)h Fo(j)i Fw(the)f(latitude)e(axis.)1032 2506 y(The)25 b(\\north-celestial-p)q (ole")e(co)q(ordinate)h(represen)o(tation)970 2556 y(established)d(b)o (y)e(Greisen)i(\(1983)o(\))f(is)f(not)h(one)g(of)f(the)i(pro)r(jec-)970 2605 y(tions)e(prop)q(osed)g(ab)q(o)o(v)o(e,)f(should)g(no)h(longer)f (b)q(e)h(written,)g(and)970 2655 y(should)c(b)q(e)h(translated)g(b)o(y) f(mo)q(dern)g(FITS-reading)g(programs.)p eop %%Page: 26 26 26 25 bop -116 -192 a Fs(26)430 b(Greisen)14 b(&)e(Calabretta:)i (Represen)o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 -93 y Fw(There)d(are)g(t)o(w)o(o)g(p)q(ossible)f(translations)h(of)f (this)h(old)e(system.)h(The)-116 -43 y(simpler)f(regards)j(the)g Fp('-NCP')e Fw(system)h(as)g(a)f(slan)o(t)h(orthographic)-116 7 y(pro)r(jection.)i(In)h(this)g(case,)h(if)e(the)i(longitude)e(and)h (latitude)f(axis)-116 57 y(n)o(um)o(b)q(ers)f(are)h Fo(m;)7 b(n)p Fw(,)k(resp)q(ectiv)o(ely)m(,)i(then)g(the)g(parameters)g(to)f(b) q(e)-116 106 y(translated)i(and)g(the)g(new)g(parameters)g(are)-116 200 y Fp(CTYPE)p Fr(m)19 b Fw(=)i Fp('RA---SIN')-106 263 y(CTYPE)p Fr(n)f Fw(=)h Fp('DEC--SIN')508 b Fw(\(134\))-104 325 y Fp(PROJP1)19 b Fw(=)i(0)-104 387 y Fp(PROJP2)e Fw(=)i(cot)8 b Fo(\016)181 393 y Fi(N)s(C)r(P)-116 481 y Fw(One)23 b(ma)o(y)d(also)i(regard)h(the)f Fp('-NCP')g Fw(system)g(as)g(an)g(ortho-)-116 531 y(graphic)16 b(pro)r(jection)h (with)g(the)g(north)g(equatorial)f(p)q(ole)g(as)h(the)-116 581 y(tangen)o(t)e(p)q(oin)o(t,)e(but)i(with)g(the)g(cen)o(ter)i(of)d (the)h(observ)n(ation)g(de-)-116 631 y(clared)f(to)h(b)q(e)g(the)g (reference)h(p)q(oin)o(t.)e(F)m(rom)e(this)j(p)q(oin)o(t)f(of)g(view,) -116 680 y(the)g(parameters)g(whic)o(h)g(need)h(to)e(b)q(e)i (translated)f(are)-12 774 y Fp(CTYPE)p Fr(m)20 b Fw(=)h Fp('RA---SIN')-1 837 y(CTYPE)p Fr(n)f Fw(=)h Fp('DEC--SIN')-1 899 y(CRVAL)p Fr(n)f Fw(=)h(90)7 b(sign)o(\()p Fo(\016)359 905 y Fi(N)s(C)r(P)442 899 y Fw(\))-12 983 y Fp(CRPIX)p Fr(m)20 b Fw(=)h Fo(m)241 989 y Fi(N)s(C)r(P)333 983 y Fw(+)380 955 y(180)443 939 y Fj(\016)p 380 973 82 2 v 408 1011 a Fo(\031)473 983 y Fw(cot)7 b Fo(\016)553 989 y Fi(N)s(C)r(P)644 983 y Fw(sin)f Fo(\032=)p Fw(\001)778 989 y Fi(m)-1 1084 y Fp(CRPIX)p Fr(n)20 b Fw(=)h Fo(n)230 1090 y Fi(N)s(C)r(P)322 1084 y Fw(+)368 1056 y(180)431 1041 y Fj(\016)p 368 1075 V 396 1113 a Fo(\031)461 1084 y Fw(cot)8 b Fo(\016)542 1090 y Fi(N)s(C)r(P)632 1084 y Fw(cos)g Fo(\032=)p Fw(\001)772 1090 y Fi(n)828 1084 y Fw(\(135\))-116 1155 y Fp(PC)p Fr(mmmmmm)20 b Fw(=)h(cos)7 b Fo(\032)-85 1217 y Fp(PC)p Fr(mmmnnn)22 b Fw(=)f Fn(\000)p Fw(\001)272 1223 y Fi(n)301 1217 y Fw(sin)7 b Fo(\032=)p Fw(\001)436 1223 y Fi(m)-85 1279 y Fp(PC)p Fr(nnnmmm)22 b Fw(=)f(\001)240 1285 y Fi(m)278 1279 y Fw(sin)7 b Fo(\032)g Fn(j)g Fw(sin)f Fo(\016)458 1285 y Fi(N)s(C)r(P)542 1279 y Fn(j)h Fo(=)p Fw(\001)617 1285 y Fi(n)-53 1342 y Fp(PC)p Fr(nnnnnn)23 b Fw(=)e(cos)7 b Fo(\032)g Fn(j)g Fw(sin)g Fo(\016)390 1348 y Fi(N)s(C)r(P)473 1342 y Fn(j)-12 1404 y Fp(CDELT)p Fr(m)20 b Fw(=)h(\001)240 1410 y Fi(m)-1 1466 y Fp(CDELT)p Fr(n)f Fw(=)h(\001)240 1472 y Fi(n)269 1466 y Fo(;)-116 1560 y Fw(where)12 b Fo(m)37 1566 y Fi(N)s(C)r(P)121 1560 y Fo(;)7 b(n)165 1566 y Fi(N)s(C)r(P)258 1560 y Fw(\(old)k(v)n(alues)h(for)f(k)o(eyw)o(ords)h Fp(CRPIX)p Fr(m)e Fw(and)-116 1610 y Fp(CRPIX)p Fr(n)p Fw(\))19 b(are)i(the)g(original)d(pixel)i(co)q(ordinates)h(of)e(the)i (refer-)-116 1660 y(ence)16 b(p)q(oin)o(t)e(on)g(the)i(longitude)e(and) g(latitude)h(\(righ)o(t)f(ascension)-116 1710 y(and)19 b(declination\))f(axes,)h Fo(\013)338 1716 y Fi(N)s(C)r(P)421 1710 y Fo(;)7 b(\016)458 1716 y Fi(N)s(C)r(P)560 1710 y Fw(\(old)18 b(v)n(alues)h(for)g(k)o(ey-)-116 1759 y(w)o(ords)d Fp(CRVAL)p Fr(m)e Fw(and)i Fp(CRVAL)p Fr(n)p Fw(\))f(are)h(the)g(co)q (ordinate)h(v)n(alues)e(at)-116 1809 y(the)f(reference)i(p)q(oin)o(t,)d (and)g(\001)364 1815 y Fi(m)395 1809 y Fo(;)7 b Fw(\001)449 1815 y Fi(n)485 1809 y Fw(\(old)13 b(v)n(alues)g(for)h(k)o(eyw)o(ords) -116 1859 y Fp(CDELT)p Fr(m)j Fw(and)h Fp(CDELT)p Fr(n)p Fw(\))f(are)i(the)g(incremen)o(ts)f(on)g(those)h(axes,)-116 1909 y Fo(\032)13 b Fw(=)i(the)g(old)f(v)n(alue)h(for)f(the)h(old)f(k)o (eyw)o(ord)h Fp(CROTA)p Fr(n)p Fw(.)f(Note)h(that)-116 1959 y(this)e(translation)g(requires)i(the)g Fp(PC)e Fw(nomenclature)g(when)h Fo(\032)e Fn(6)p Fw(=)g(0)-116 2008 y(since)17 b(the)h(rotation)e(is)h(o\013set)h(from)e(the)h(tangen) o(t)g(p)q(oin)o(t.)f(Note)-116 2058 y(that,)f(when)i Fo(\032)f Fw(=)g(0,)f(one)i(can)f(also)g(set)h(the)g Fp(PC)f Fw(matrix)e(to)i(the)-116 2108 y(unit)o(y)d(matrix)f(and)i Fp(CDELT)p Fr(n)f Fw(to)h(\001)440 2114 y Fi(n)469 2108 y Fw(sin)7 b Fo(\016)545 2114 y Fi(N)s(C)r(P)628 2108 y Fw(.)-116 2227 y Fx(6.)16 b(T)l(utorial)d(discussion)-54 2307 y Fw(The)k(material)d(so)j(far)f(presen)o(ted)j(has)d(b)q(een)i (quite)f(general)-116 2356 y(and)g(all)g(encompassing.)f(F)m(or)h (tutorial)g(purp)q(oses,)i(let)f(us)g(con-)-116 2406 y(sider)c(a)g(concrete)h(example)e(of)g(a)h(\\normal")d(optical)i (image.)e(Its)-116 2456 y(header)21 b(could)e(lo)q(ok)g(lik)o(e)h(T)m (able)11 b(2.)20 b(The)g Fp(NAXIS)f Fw(and)h Fp(NAXIS)p Fr(n)-116 2506 y Fw(k)o(eyw)o(ords)e(tell)f(us)h(that)g(w)o(e)g(ha)o(v) o(e)g(a)g(four-dimensional)d(image)-116 2556 y(consisting)g(of)f(512)g (columns,)f(512)h(ro)o(ws,)h(196)f(planes,)h(and)g(one)-116 2605 y(cub)q(e.)d(The)h(last)f(axis)f(is)h(used)h(simply)d(to)i(con)o (v)o(ey)g(another)h(co)q(or-)-116 2655 y(dinate)g(v)n(alue)f(ev)o(en)i (though)f(the)h(en)o(tire)f(image)f(has)h(that)g(v)n(alue.)970 -93 y(One-p)q(oin)o(t)19 b(axes)f(should)h(b)q(e)g(put)f(at)g(the)h (end)g(of)f(the)h(list)f(for)970 -43 y(simplicit)o(y)m(,)f(but)j(they)g (are)h(allo)o(w)o(ed)d(to)i(b)q(e)h(an)o(ywhere)f(within)970 7 y(the)e(list.)e(The)h Fp(CRPIX)p Fr(n)g Fw(k)o(eyw)o(ords)g(tell)g (us)g(that)g(the)h(reference)970 63 y(p)q(oin)o(t)g(is)f(at)h(the)h (cen)o(ter)g(of)e(the)i(pixel)e(in)h(the)g(256)1808 48 y(th)1867 63 y(column,)970 119 y(257)1033 104 y(th)1086 119 y(ro)o(w,)11 b(\014rst)h(plane,)g(and)f(\014rst)i(cub)q(e.)g(The)f Fp(PC)f Fw(k)o(eyw)o(ords)i(all)970 169 y(ha)o(v)o(e)i(their)g(default) f(v)n(alues)h(implying)c(no)k(rotation)f(or)h(sk)o(ew)g(in)970 219 y(the)g(co)q(ordinates.)1032 300 y(Th)o(us,)d(the)h(relativ)o(e)f (co)q(ordinate)h(v)o(ector)g(ma)o(y)d(b)q(e)j(computed)970 350 y(as)19 b(sho)o(wn)g(in)g(T)m(able)12 b(3)18 b(from)g(Eq.)12 b(3.)18 b(Since)i Fp('VELOCITY')d Fw(and)970 400 y Fp('STOKES')e Fw(are)h(linear)g(axes,)g(the)h(actual)e(v)o(elo)q(cit)o(y)h(and)g (Stok)o(es)970 449 y(of)f(eac)o(h)h(p)q(oin)o(t)g(are)g(found)f(simply) f(b)o(y)h(adding)g(the)h(co)q(ordinate)970 499 y(v)n(alue)970 549 y(at)k(the)g(reference)i(p)q(oin)o(t)d(to)h(the)g(relativ)o(e)f(co) q(ordinate.)h(Th)o(us,)970 596 y(V)m(elo)q(cit)o(y)49 b(=)i(500000)p Fo(:)7 b Fw(+)i(7128)p Fo(:)p Fw(3\()p Fo(k)g Fn(\000)g Fw(1\))50 b(meters/sec)970 646 y(Stok)o(es)81 b(=)51 b(1)p Fo(:)p Fw(0)8 b(+)h(\()p Fo(l)i Fn(\000)e Fw(1\))258 b(=)14 b(I)g(p)q(olariz.)1032 725 y(The)28 b(spherical)f(co)q(ordinates)h(are)f(not)g(as)g(simple.)e(The)970 775 y Fp(CTYPE)p Fr(n)20 b Fw(k)o(eyw)o(ords)h(tell)f(us)h(that)g(the)g (pro)r(jection)g(is)g(of)f(t)o(yp)q(e)970 825 y Fp(TAN)f Fw(whic)o(h)g(is)g(zenithal)g(pro)r(jection)g(with)g(the)h(source)g(of) f(the)970 875 y(pro)r(jection)13 b(at)e(the)i(cen)o(ter)g(of)f(the)g (sphere.)h(The)g(reference)h(p)q(oin)o(t)970 924 y(is)g(the)h(tangen)o (t)f(p)q(oin)o(t)g(whic)o(h)g(is)g(the)h(north)f(p)q(ole)g(of)g(the)h (nativ)o(e)970 974 y(co)q(ordinate)g(system.)f(The)g(nativ)o(e)g (longitude)g(and)g(latitude)g(are)970 1024 y(then)h(giv)o(en)e(b)o(y) 970 1178 y Fo(\036)21 b Fw(=)g(arg)7 b(\()p Fn(\000)p Fo(y)q(;)g(x)p Fw(\))974 1278 y Fo(\022)23 b Fw(=)e(tan)1129 1261 y Fj(\000)p Fu(1)1180 1207 y Fl( )1218 1250 y Fw(180)1281 1235 y Fj(\016)p 1218 1268 V 1246 1306 a Fo(\031)1386 1250 y Fw(1)p 1309 1268 176 2 v 1309 1277 a Fl(p)p 1351 1277 134 2 v 36 x Fo(x)1375 1301 y Fu(2)1402 1313 y Fw(+)10 b Fo(y)1465 1301 y Fu(2)1489 1207 y Fl(!)1536 1278 y Fo(;)970 1407 y Fw(whic)o(h,)j(on)h(substitution,)g(b)q(ecome)970 1473 y Fo(\036)21 b Fw(=)g(arg)7 b(\()p Fo(j)12 b Fn(\000)d Fw(257)p Fo(;)e(i)i Fn(\000)g Fw(256\))g(+)g(180)1558 1456 y Fj(\016)974 1573 y Fo(\022)23 b Fw(=)e(tan)1129 1555 y Fj(\000)p Fu(1)1180 1502 y Fl( )1335 1545 y Fw(20626)p Fo(:)p Fw(48062)p 1218 1563 454 2 v 1218 1571 a Fl(p)p 1260 1571 413 2 v 36 x Fw(\()p Fo(i)9 b Fn(\000)h Fw(256\))1420 1595 y Fu(2)1447 1607 y Fw(+)f(\()p Fo(j)j Fn(\000)e Fw(257\))1654 1595 y Fu(2)1677 1502 y Fl(!)1724 1573 y Fo(;)970 1762 y Fw(using)k(Eqs.)f(136,)g(12,)g(17)h(and)g(13.)g(The)h (recognized)g(co)q(ordinate)970 1812 y(system)h(is)f(equatorial)g (since)h(the)g Fp(CTYPE)p Fr(n)f Fw(b)q(egin)g(with)h Fp(RA)f Fw(and)970 1861 y Fp(DEC)20 b Fw(and)h(the)h(reference)h(p)q (oin)o(t)e(has)g(equatorial)f(co)q(ordinates)970 1911 y(45)p Fo(:)1024 1896 y Fj(\016)1042 1911 y Fw(83)d(in)g(righ)o(t)g (ascension)h(and)f(63)p Fo(:)1581 1896 y Fj(\016)1599 1911 y Fw(57)g(in)g(declination)f(from)970 1961 y(the)f Fp(CRVAL)p Fr(n)p Fw(.)e(The)i(equatorial)e(north)i(p)q(ole)f(has)h(a)f (longitude)f(of)970 2011 y(180)1033 1996 y Fj(\016)1067 2011 y Fw(in)i(the)h(nativ)o(e)g(co)q(ordinate)g(system)f(from)f(the)i Fp(LONGPOLE)970 2061 y Fw(k)o(eyw)o(ord.)10 b(Since)g(this)g(is)g(a)g (zenithal)f(pro)r(jection,)h Fp(LATPOLE)f Fw(is)h(not)970 2111 y(giv)o(en.)h(Th)o(us,)g(Eqs.)h(6)g(for)f(the)h(righ)o(t)f (ascension)h(and)f(declination)970 2160 y(b)q(ecome)970 2304 y(sin)c Fo(\016)14 b Fw(=)d(sin)c Fo(\022)i Fw(sin\(63)p Fo(:)p Fw(57\))f Fn(\000)h Fw(cos)f Fo(\022)g Fw(cos)g Fo(\036)f Fw(cos\(63)p Fo(:)p Fw(57\))970 2354 y(cos)h Fo(\016)h Fw(sin\()p Fo(\013)g Fn(\000)g Fw(45)p Fo(:)p Fw(83\))h(=)i(cos)c Fo(\022)g Fw(sin)f Fo(\036)970 2403 y Fw(cos)h Fo(\016)h Fw(cos\()p Fo(\013)g Fn(\000)h Fw(45)p Fo(:)p Fw(83\))g(=)i(sin)7 b Fo(\022)h Fw(cos)q(\(63)p Fo(:)p Fw(57\))1373 2453 y(+)h(cos)f Fo(\022)g Fw(cos)g Fo(\036)f Fw(sin)o(\(63)p Fo(:)p Fw(57\))g Fo(:)970 2605 y Fw(Therefore)20 b(the)g(co)q(ordinates)f(of,)f(for)g(example,)f(the)i (\014rst)h(and)970 2655 y(last)14 b(corners)h(of)f(the)g(image)e(are)p eop %%Page: 27 27 27 26 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(27)-116 -67 y Fm(T)l(able)14 b(2.)f Fs(Example)i(FITS)d(header)i (with)g(co)q(ordinates)210 24 y Fp(NAXIS)87 b(=)413 b(4)131 b(/)22 b(4-dimensional)d(cube)210 74 y(NAXIS1)65 b(=)370 b(512)130 b(/)22 b(x)f(axis)g(\(fastest\))210 124 y(NAXIS2)65 b(=)370 b(512)130 b(/)22 b(y)f(axis)g(\(2nd)g(fastest\))210 174 y(NAXIS3)65 b(=)370 b(196)130 b(/)22 b(z)f(axis)g(\(planes\))210 224 y(NAXIS4)65 b(=)413 b(1)131 b(/)22 b(dummy)e(to)i(give)f(a)g (coordinate)210 274 y(CTYPE1)65 b(=)21 b('RA---TAN')325 b(/)22 b(TAN)f(projection)e(used)210 323 y(CRVAL1)65 b(=)326 b(45.83)130 b(/)22 b(RA)f(at)g(reference)f(point)210 373 y(CRPIX1)65 b(=)370 b(256)130 b(/)22 b(pixel)e(i)i(of)f(reference)f (point)210 423 y(CDELT1)65 b(=)196 b(-0.00277777)128 b(/)22 b(10)f(arcsec)g(per)g(pixel)210 473 y(PC001001)g(=)370 b(1.0)130 b(/)22 b(no)f(rotation,)f(skew)210 523 y(CTYPE2)65 b(=)21 b('DEC--TAN')325 b(/)22 b(TAN)f(projection)e(used)210 572 y(CRVAL2)65 b(=)326 b(63.57)130 b(/)22 b(Dec)f(at)g(reference)f (point)210 622 y(CRPIX2)65 b(=)370 b(257)130 b(/)22 b(pixel)e(j)i(of)f (reference)f(point)210 672 y(CDELT2)65 b(=)217 b(0.00277777)129 b(/)22 b(10)f(arcsec)g(per)g(pixel)210 722 y(PC002002)g(=)370 b(1.0)130 b(/)22 b(no)f(rotation,)f(skew)210 772 y(CTYPE3)65 b(=)21 b('VELOCITY')325 b(/)22 b(each)f(plane)f(at)i(a)f(velocity)210 821 y(CRVAL3)65 b(=)261 b(500000.0)129 b(/)22 b(velocity)e(in)h(m/sec) 210 871 y(CRPIX1)65 b(=)413 b(1)131 b(/)22 b(pixel)e(k)i(of)f (reference)f(point)210 921 y(CDELT3)65 b(=)304 b(7128.3)130 b(/)22 b(7128.3)e(m/sec)210 971 y(PC003003)h(=)370 b(1.0)130 b(/)22 b(no)f(rotation,)f(skew)210 1021 y(CTYPE4)65 b(=)21 b('STOKES)42 b(')327 b(/)22 b(Polarization)210 1071 y(CRVAL4)65 b(=)413 b(1)131 b(/)22 b(unpolarized)210 1120 y(CRPIX4)65 b(=)413 b(1)131 b(/)22 b(at)f(our)g(dummy)g(plane)210 1170 y(CDELT4)65 b(=)413 b(1)131 b(/)22 b(or)f(anything)f(here.)210 1220 y(PC004004)h(=)370 b(1.0)130 b(/)22 b(no)f(rotation,)f(skew)210 1270 y(LONGPOLE)h(=)370 b(180)130 b(/)22 b(native)e(longitude)g(of)h (equatorial)f(pole)-116 1438 y Fm(T)l(able)14 b(3.)f Fs(Example)i(FITS)d(header)i(co)q(ordinate)h(computation)61 1560 y Fl(0)61 1633 y(B)61 1658 y(B)61 1684 y(@)97 1593 y Fo(x)98 1643 y(y)98 1693 y(z)99 1742 y(s)131 1560 y Fl(1)131 1633 y(C)131 1658 y(C)131 1684 y(A)188 1668 y Fw(=)241 1560 y Fl(0)241 1633 y(B)241 1658 y(B)241 1684 y(@)278 1593 y Fp(CDELT1)74 b(0)129 b(0)h(0)332 1643 y(0)75 b(CDELT2)f(0)130 b(0)332 1693 y(0)g(0)75 b(CDELT3)f(0)332 1742 y(0)130 b(0)f(0)75 b(CDELT4)873 1560 y Fl(1)873 1633 y(C)873 1658 y(C)873 1684 y(A)923 1560 y(0)923 1633 y(B)923 1658 y(B)923 1684 y(@)960 1593 y Fp(PC001001)19 b(PC001002)g(PC001003)g(PC001004)960 1643 y(PC002001)g(PC002002)g(PC002003)g(PC002004)960 1693 y(PC003001)g(PC003002)g(PC003003)g(PC003004)960 1742 y(PC004001)g(PC004002)g(PC004003)g(PC004004)1730 1560 y Fl(1)1730 1633 y(C)1730 1658 y(C)1730 1684 y(A)1773 1560 y(0)1773 1633 y(B)1773 1658 y(B)1773 1684 y(@)1817 1593 y Fo(i)10 b Fn(\000)f Fo(i)1896 1599 y Fu(0)1813 1643 y Fo(j)j Fn(\000)e Fo(j)1901 1649 y Fu(0)1809 1693 y Fo(k)g Fn(\000)g Fo(k)1905 1699 y Fu(0)1819 1742 y Fo(l)g Fn(\000)g Fo(l)1895 1748 y Fu(0)1934 1560 y Fl(1)1934 1633 y(C)1934 1658 y(C)1934 1684 y(A)-116 1803 y Fw(whic)o(h)j(b)q (ecomes)61 1829 y Fl(0)61 1902 y(B)61 1927 y(B)61 1954 y(@)97 1863 y Fo(x)98 1912 y(y)98 1962 y(z)99 2012 y(s)131 1829 y Fl(1)131 1902 y(C)131 1927 y(C)131 1954 y(A)188 1938 y Fw(=)241 1829 y Fl(0)241 1902 y(B)241 1927 y(B)241 1954 y(@)278 1863 y Fn(\000)p Fw(0)p Fo(:)p Fw(00277777)107 b(0)156 b(0)68 b(0)383 1912 y(0)125 b(0)p Fo(:)p Fw(00277777)65 b(0)j(0)383 1962 y(0)214 b(0)109 b(7128)p Fo(:)p Fw(3)19 b(0)383 2012 y(0)214 b(0)156 b(0)68 b(1)915 1829 y Fl(1)915 1902 y(C)915 1927 y(C)915 1954 y(A)958 1829 y(0)958 1902 y(B)958 1927 y(B)958 1954 y(@)997 1863 y Fo(i)10 b Fn(\000)f Fw(256)995 1912 y Fo(j)j Fn(\000)d Fw(257)1014 1962 y Fo(k)h Fn(\000)g Fw(1)1019 2012 y Fo(l)g Fn(\000)g Fw(1)1138 1829 y Fl(1)1138 1902 y(C)1138 1927 y(C)1138 1954 y(A)1188 1938 y Fo(:)715 b Fw(\(136\))-65 2191 y(parameter)p -65 2205 185 2 v 50 w(units)p 170 2198 91 2 v 91 w(\014rst)15 b(corner)p 351 2198 199 2 v 106 w(last)e(corner)p 655 2198 192 2 v -15 2242 a(\()p Fo(i;)7 b(j)r Fw(\))101 b(pixels)113 b(\(0.5,)13 b(0.5\))49 b(\(512.5,)12 b(512.5\))-16 2292 y(\()p Fo(k)q(;)7 b(l)q Fw(\))99 b(pixels)113 b(\(0.5,)13 b(1.0\))90 b(\(196.5,)12 b(1.0\))15 2341 y Fo(\036)130 b Fw(degrees)135 b(44.888)159 b(225.112)17 2391 y Fo(\022)134 b Fw(degrees)52 b(88.9944412)95 b(88.9944412)14 2441 y Fo(\013)129 b Fw(degrees)52 b(47.3852040)95 b(44.1887934)17 2491 y Fo(\016)135 b Fw(degrees)52 b(62.8489681)95 b(64.2704912)-46 2541 y(V)m(elo)q(cit)o(y)69 b(m/s)129 b(496435.85)96 b(1893582.65)-30 2591 y(Stok)o(es)85 b(t)o(yp)q(e)175 b(1)p Fo(:)p Fw(0)11 b Fn(\021)j Fw(I)171 b(1)p Fo(:)p Fw(0)10 b Fn(\021)15 b Fw(I)970 2197 y(Ev)o(en)f(in)e(this)h(large)g (\()p Fn(\030)f Fw(2)1383 2182 y Fj(\016)1415 2197 y Fw(diagonal\))f(image,)f(the)k(e\013ect)h(of)d(the)970 2247 y(pro)r(jection)i(non-linearit)o(y)e(is)i(only)e(0.04)h(pixels)g (at)g(the)h(corners.)970 2297 y(If)d(the)i(pixel)e(separation)g(w)o (ere)i(ten)f(times)f(larger,)g(then)h(the)h(pro-)970 2347 y(jection)20 b(non-linearit)o(y)e(w)o(ould)h(b)q(e)i(3.4)d(pixels) i(at)g(the)g(corners)970 2396 y(whic)o(h)14 b(is)f(a)g(thousand)h (times)e(larger.)h(The)h(righ)o(t)f(ascension)i(o\013-)970 2446 y(set)i(from)c(the)j(reference)i(p)q(oin)o(t)d(is)h(a\013ected)h (b)o(y)e(the)h(cos)8 b Fo(\016)17 b Fw(term)970 2496 y(and)d(is,)g(therefore,)i(quite)e(nonlinear)g(ev)o(en)h(in)f(this)h (fairly)e(small)970 2546 y(image.)h(The)j(t)o(w)o(o)e Fn(j)p Fo(\013)10 b Fn(\000)h Fo(\013)1391 2552 y Fi(P)1419 2546 y Fn(j)k Fw(di\013er)i(b)o(y)f(20.66)f(seconds)i(of)f(time)970 2596 y(\(309.8)11 b(arc)i(seconds\).)h(This)f(co)q(ordinate)g(rotation) f(nonlinearit)o(y)p eop %%Page: 28 28 28 27 bop -116 -192 a Fs(28)430 b(Greisen)14 b(&)e(Calabretta:)i (Represen)o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 -93 y Fw(is)d(small)f(for)h(reference)k(p)q(oin)o(ts)c(near)i(the)f (equator)g(and)g(large)f(for)-116 -43 y(p)q(oin)o(ts)i(near)i(the)f(p)q (oles.)-54 43 y(It)19 b(is)f(clear)i(that)f(non-linearities)f(are)h(v)o (ery)g(imp)q(ortan)o(t)e(in)-116 93 y(the)e(all-sky)f(and)g(other)i (large-\014eld)e(pro)r(jections,)h(but)g(it)g(is)f(less)-116 142 y(clear)j(that)g(the)g(pro)r(jectiv)o(e)h(non-linearities)e(matter) h(for)f(small)-116 192 y(\014elds.)d(The)i Fp(SIN)e Fw(pro)r(jection)h (alw)o(a)o(ys)f(has)h(an)g Fo(R)660 198 y Fi(\022)692 192 y Fw(less)g(than)g(the)-116 242 y(the)19 b(other)h(zenithal)f(pro)r (jections.)g(Therefore,)h(w)o(e)f(ha)o(v)o(e)g(plot-)-116 292 y(ted)c(the)h(di\013erence)h(b)q(et)o(w)o(een)f(the)g Fo(R)482 298 y Fi(\022)515 292 y Fw(for)f(v)n(arious)f(pro)r(jections) -116 342 y(and)i(that)h(for)g(the)g Fp(SIN)g Fw(pro)r(jection)g(as)g(a) f(function)h(of)f(natural)-116 391 y(latitude)h Fo(\022)j Fw(in)e(Fig.)11 b(30)18 b(with)g(the)h(small)c(angles)k(displa)o(y)o (ed)e(on)-116 441 y(a)d(di\013eren)o(t)j(scale)e(in)g(Fig.)c(31.)j(W)m (e)h(note)h(that)f(the)h Fp(TAN)e Fw(pro)r(jec-)-116 491 y(tion)e(blo)o(ws)h(up)h(at)f Fo(\022)g Fw(=)f(0,)h(while)g(the)h (di\013erences)h(for)e(the)h(other)-116 541 y(pro)r(jections)g(approac) o(h)f(some)g(10's)g(of)g(degrees.)i(The)f(di\013erence)-116 591 y(for)i(all)g(pro)r(jections)h(exceeds)i(1)e(arc)g(second)h(for)f Fo(\022)h(<)f Fw(88)821 576 y Fj(\016)856 591 y Fw(and)-116 641 y(the)c(di\013erence)j(for)d(the)h Fp(TAN)e Fw(pro)r(jection)i (exceeds)h(one)f(millia)o(rc-)-116 690 y(second)i(only)f(440)g(arc)h (seconds)i(from)c(the)i(natural)f(p)q(ole.)g(Suc)o(h)-116 740 y(nonlinearities)10 b(w)o(ould)g(b)q(e)i(signi\014can)o(t)f(in)f (radio)h(VLB)g(and)g(other)-116 790 y(v)o(ery-high)i(resolution)h (observ)n(ations.)-54 876 y(The)d(role)g(of)f(the)i(co)q(ordinate)f(v)n (alues)g(at)f(the)i(reference)h(p)q(oin)o(t)-116 926 y(and)19 b(of)f Fp(LONGPOLE)g Fw(dep)q(ends)j(on)e(the)g(sort)h(of)f (imaging)d(b)q(eing)-116 975 y(done.)i(The)h(zenithal)g(pro)r(jections) h(called)e Fp(TAN)p Fw(,)g Fp(ARC)g Fw(and)h Fp(SIN)-116 1025 y Fw(are)13 b(the)h(natural)f(result)h(of)e(certain)i(tec)o (hniques)h(of)d(observ)n(ation)-116 1075 y(and)k(the)i(co)q(ordinate)f (v)n(alues)g(at)g(the)g(reference)i(p)q(oin)o(t)e(are)g(the)-116 1125 y(tangen)o(t)c(p)q(oin)o(t)f(of)h(those)h(observ)n(ations)f (\(usually)f(the)i(telescop)q(e)-116 1175 y(p)q(oin)o(ting)j(p)q (osition\).)g(The)h(longitude)g(of)f(the)i(p)q(ole)f(parameter)-116 1224 y(then)g(con)o(v)o(eys)g(an)o(y)g(rotation)f(of)g(the)i(scene)g (whic)o(h)f(the)g(imag-)-116 1274 y(ing)12 b(instrumen)o(t)i(ma)o(y)d (ha)o(v)o(e)j(caused.)g(Geometries)f(whic)o(h)g(co)o(v)o(er)-116 1324 y(large)j(areas)g(of)g(the)h(sky)g(ha)o(v)o(e)f(some)f(regions)i (whic)o(h)f(are)h(little)-116 1374 y(distorted)j(in)f(either)i(area,)e (angles,)g(or)h(shap)q(e)g(and)g(other)g(ar-)-116 1424 y(eas)f(whic)o(h)f(are)h(highly)f(distorted)h(in)f(t)o(w)o(o)h(or)f (more)g(of)g(these.)-116 1474 y(By)13 b(selecting)h(the)g(reference)i (p)q(oin)o(t)d(lo)q(cation)f(and)i(the)g(rotation)-116 1523 y(of)h(the)h(scene,)h(one)f(can)g(alter)g(whic)o(h)g(p)q(ortions)g (of)f(the)i(sky)e(are)-116 1573 y(highly)d(distorted)i(and)f(whic)o(h)h (are)f(represen)o(ted)k(fairly)12 b(reliably)m(.)-116 1623 y(Figure)g(32)18 b(sho)o(ws)g(the)h(zenithal)f(equal)g(area)g(pro) r(jection)g(with)-116 1673 y(four)f(di\013eren)o(t)h(c)o(hoices)h(of)e (reference)j(parameters.)d(The)h(same)-116 1723 y(parameter)e(c)o (hoices)i(are)f(illustrated)f(for)h(the)g(conical)f(equidis-)-116 1772 y(tan)o(t)f(pro)r(jection)i(in)e(Fig.)d(33,)j(the)h (Hammer-Aito\013)d(pro)r(jection)-116 1822 y(in)k(Fig.)11 b(34,)16 b(and)i(Cob)q(e's)f(Quadrilateralized)h(Spherical)f(Cub)q(e) -116 1872 y(in)c(Fig.)e(35.)-54 1958 y(In)16 b(writing)f(computer)g (programs)g(to)h(handle)g(co)q(ordinates,)-116 2008 y(it)f(is)g (relativ)o(ely)g(simple)f(to)i(write)g(ones)g(whic)o(h)g(do)f(the)i (con)o(v)o(er-)-116 2057 y(sions)c(from)e(pixel)i(n)o(um)o(b)q(ers)g (\()p Fo(i;)7 b(j)r Fw(\))13 b(to)g(standard)h(celestial)f(\()p Fo(\013;)7 b(\016)r Fw(\))-116 2107 y(co)q(ordinates)13 b(and)f(the)h(in)o(v)o(erse)g(using)f(the)h(four)f(steps)i(describ)q (ed)-116 2157 y(in)g(Section)f(4.)h(Ho)o(w)o(ev)o(er,)h(in)g(a)f(v)n (ariet)o(y)h(of)f(circumstances)i(suc)o(h)-116 2207 y(as)f(the)h (plotting)f(of)g(tic)o(k)g(marks)g(on)g(image)e(displa)o(ys,)i(the)h (pro-)-116 2257 y(gram)e(kno)o(ws)j Fo(i)f Fw(and)g Fo(\016)r Fw(,)g(for)g(example,)f(and)h(requires)i Fo(j)g Fw(and)f Fo(\013)p Fw(.)-116 2307 y(F)m(or)10 b(suc)o(h)i(cases,)h(it)d(w)o (ould)h(b)q(e)h(con)o(v)o(enien)o(t)f(to)g(com)o(bine)f(the)i(four)-116 2356 y(steps)g(in)o(to)f(t)o(w)o(o)g(equations)h(in)f(the)h(four)f(v)n (ariables,)f(so)i(that)f(they)-116 2406 y(ma)o(y)d(b)q(e)j(solv)o(ed)f (for)g(an)o(y)g(t)o(w)o(o)g(giv)o(en)g(the)h(other)g(t)o(w)o(o.)e(F)m (or)h(some)f(of)-116 2456 y(the)14 b(pro)r(jections,)g(this)f(ma)o(y)f (b)q(e)i(done)g(analytically)m(,)c(but)k(others)-116 2506 y(are)g(not)h(so)g(simple.)d(W)m(e)i(presen)o(t)i(some)e(of)g(the) h(details)f(of)g(suc)o(h)-116 2556 y(computations)c(in)h(App)q(endix)i (C.)e(F)m(or)g(the)h(more)f(di\016cult)g(cases,)-116 2605 y(iterativ)o(e)20 b(tec)o(hniques)i(will)d(b)q(e)i(needed.)g(W)m (e)f(remind)f(readers)-116 2655 y(that)12 b(the)h(simplest)e(p)q(o)o(w) o(erful)g(iterativ)o(e)h(tec)o(hnique)i(is)e(Newton's)970 -93 y(Metho)q(d,)i(namely)m(,)d(at)j(the)h Fo(i)9 b Fw(+)h(1)j (iteration,)1012 18 y Fo(x)1036 24 y Fi(i)p Fu(+1)1112 18 y Fw(=)21 b Fo(x)1189 24 y Fi(i)1212 18 y Fn(\000)1253 -41 y Fl(\024)1298 -11 y Fo(f)t(g)1342 -5 y Fi(y)1372 -11 y Fn(\000)9 b Fo(g)q(f)1454 -5 y Fi(y)p 1280 8 213 2 v 1280 46 a Fo(f)1300 52 y Fi(x)1321 46 y Fo(g)1341 52 y Fi(y)1370 46 y Fn(\000)h Fo(g)1432 52 y Fi(x)1453 46 y Fo(f)1473 52 y Fi(y)1498 -41 y Fl(\025)1520 60 y Fu(\()p Fi(x)1552 64 y Fb(i)1565 60 y Fi(;y)1592 64 y Fb(i)1605 60 y Fu(\))1015 144 y Fo(y)1035 150 y Fi(i)p Fu(+1)1112 144 y Fw(=)21 b Fo(y)1185 150 y Fi(i)1208 144 y Fn(\000)1250 85 y Fl(\024)1294 116 y Fo(g)q(f)1335 122 y Fi(x)1366 116 y Fn(\000)9 b Fo(f)t(g)1451 122 y Fi(x)p 1277 134 V 1277 172 a Fo(f)1297 178 y Fi(x)1318 172 y Fo(g)1338 178 y Fi(y)1367 172 y Fn(\000)h Fo(g)1429 178 y Fi(x)1449 172 y Fo(f)1469 178 y Fi(y)1495 85 y Fl(\025)1516 186 y Fu(\()p Fi(x)1548 190 y Fb(i)1561 186 y Fi(;y)1588 190 y Fb(i)1601 186 y Fu(\))1630 144 y Fo(;)273 b Fw(\(137\))970 239 y(where)15 b(w)o(e)f(are)h(seeking)f (the)g(ro)q(ots)h(of)970 306 y Fo(f)t Fw(\()p Fo(x;)7 b(y)q Fw(\))22 b(=)f(0)974 368 y Fo(g)q Fw(\()p Fo(x;)7 b(y)q Fw(\))21 b(=)g(0)7 b Fo(;)970 456 y Fw(and)16 b(the)g Fo(x)g Fw(and)f Fo(y)j Fw(subscripts)g(refer)e(to)g(di\013eren)o (tiation)g(of)f(the)970 505 y(function)h(b)o(y)f Fo(x)g Fw(and)h Fo(y)q Fw(,)g(resp)q(ectiv)o(ely)m(.)g(In)g(one)g(dimension,)d (this)970 555 y(simpli\014es)g(to)971 665 y Fo(x)995 671 y Fi(i)p Fu(+1)1072 665 y Fw(=)21 b Fo(x)1149 671 y Fi(i)1172 665 y Fn(\000)1213 607 y Fl(\024)1248 637 y Fo(f)t Fw(\()p Fo(x)1312 643 y Fi(i)1327 637 y Fw(\))p 1240 656 112 2 v 1240 694 a Fo(f)1260 700 y Fi(x)1281 694 y Fw(\()p Fo(x)1321 700 y Fi(i)1335 694 y Fw(\))1356 607 y Fl(\025)1392 665 y Fo(;)511 b Fw(\(138\))970 750 y(where)15 b(w)o(e)f(are)h(seeking)f(the)g(ro)q(ots)h(of)970 817 y Fo(f)t Fw(\()p Fo(x)p Fw(\))22 b(=)f(0)7 b Fo(:)970 904 y Fw(Another)15 b(useful)f(relationship)f(to)h(remem)o(b)q(er)f(is) h(that,)f(if)971 992 y(0)21 b(=)g Fo(\014)10 b Fw(sin)c Fo(a)k Fw(+)f Fo(\015)h Fw(cos)d Fo(a)i Fn(\000)h Fo(Z)g(;)970 1054 y Fw(then)970 1148 y Fo(a)21 b Fw(=)g(sin)1117 1130 y Fj(\000)p Fu(1)1168 1077 y Fl( )1280 1120 y Fo(Z)p 1206 1138 179 2 v 1206 1147 a Fl(p)p 1248 1147 138 2 v 36 x Fo(\014)1273 1171 y Fu(2)1301 1183 y Fw(+)10 b Fo(\015)1366 1171 y Fu(2)1390 1077 y Fl(!)1432 1148 y Fn(\000)g Fw(arg)d(\()p Fo(\014)r(;)g(\015)r Fw(\))g Fo(:)970 1294 y Fx(7.)16 b(Summary)970 1371 y Fw(The)d(c)o(hanges)h(to) e(FITS-header)i(k)o(eyw)o(ords)f(are)g(summarized)e(in)970 1421 y(T)m(able)h(4.)e(As)i(describ)q(ed)h(in)e(Section)h(5,)f(some)f (of)h(the)g(deprecated)970 1471 y(k)o(eyw)o(ords)j(should)f(b)q(e)i (used)f(along)e(with)h(the)i(new)f(k)o(eyw)o(ords)f(so)970 1520 y(that)20 b(the)g(co)q(ordinate)h(information)16 b(ma)o(y)i(b)q(e)j(understo)q(o)q(d)g(b)o(y)970 1570 y(soft)o(w)o(are)14 b(systems)g(whic)o(h)g(ha)o(v)o(e)g(y)o(et)g(to)g (b)q(e)g(con)o(v)o(erted)h(to)f(these)970 1620 y(new)h(con)o(v)o(en)o (tions.)1032 1670 y(The)g(last)f(three)i(c)o(haracters)g(of)e Fp(CTYPE)p Fr(i)f Fw(k)o(eyw)o(ords)h(giv)o(e)g(the)970 1720 y(t)o(yp)q(e)j(of)e(pro)r(jection)i(used)g(for)f(spherical)g(co)q (ordinates.)g(The)h(3-)970 1769 y(letter)g(co)q(des)h(prop)q(osed)f(in) e(this)h(pap)q(er)h(are)g(listed)f(in)f(T)m(able)d(5.)970 1819 y(The)k(column)d(lab)q(eled)i(RP)f(giv)o(es)h(the)h(nativ)o(e)e (longitude)h(of)f(the)970 1869 y(reference)23 b(p)q(oin)o(t)e(in)f (degrees.)i(The)f(required)h(pro)r(jection)f(pa-)970 1919 y(rameters)c(are)h(listed)f(in)g(the)g(nomenclature)g(used)h (previously)m(.)970 1969 y(When)g(no)f(pro)r(jection)g(parameter)g(is)g (listed,)g(none)h(should)f(b)q(e)970 2019 y(giv)o(en.)c(Some)f(of)h (the)h(\\standard")g(pro)r(jections)g(whic)o(h)f(are)h(sim-)970 2068 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y(accoun)o(t)j(for)f(rotation)g(and)g(sk)o(ewness,)i(and)e(a)g (zenithal)h(p)q(olyno-)-116 2039 y(mial)i(pro)r(jection)j(\()p Fp(ZPN)p Fw(\))g(migh)o(t)d(b)q(e)k(used)g(to)e(accoun)o(t)i(for)e(an)o (y)-116 2088 y(radially)g(symmetric)h(comp)q(onen)o(t.)g(The)i(pixel)f (regularization)-116 2138 y(image)11 b(w)o(ould)j(then)g(b)q(e)h(used)f (to)g(correct)i(for)d(residual)h(o\013sets.)-54 2201 y(W)m(e)k(en)o(visage)h(that)f(the)h(pixel)f(regularization)g(image)e (will)-116 2251 y(b)q(e)j(used)h(in)e(an)o(y)g(of)g(three)i(w)o(a)o(ys) f(b)o(y)f(FITS-reading)h(soft)o(w)o(are)-116 2301 y(pac)o(k)n(ages:) -103 2406 y(1.)28 b(Ignore)13 b(it.)f(The)h(p)q(eak)g(and)f(RMS)g (error)h(so)g(in)o(tro)q(duced)g(will)-54 2456 y(b)q(e)g(commensurate)e (with)i(the)g(p)q(eak)f(and)h(RMS)f(error)h(of)f(the)-54 2506 y(pixel)i(regularization)g(image)f(v)n(alues.)g(\(The)j(error)f (in)f(angu-)-54 2556 y(lar)i(displacemen)o(t)h(will)f(b)q(e)h(w)o(eigh) o(ted)h(b)o(y)e(v)n(ariable)g(scaling)-54 2605 y(across)f(the)f (image\).)-103 2655 y(2.)30 b(Apply)13 b(it)h(once)g(and)g(for)g(all)e (b)o(y)i(re-gridding)f(the)i(image.)984 1378 y(3.)27 b(A)o(ttac)o(h)12 b(the)g(pixel)f(regularization)g(image)f(to)h(the)h (image)e(for)1032 1428 y(use)15 b(whenev)o(er)g(co)q(ordinate)g(con)o (v)o(ersions)f(are)g(required.)970 1490 y(This)g(third)g(usage)h (implies)d(that)i(the)h(pixel)e(regularization)h(im-)970 1540 y(age)k(m)o(ust)f(b)q(e)i(amenable)e(to)h(sub-imaging)d(op)q (erations,)j(while)970 1590 y(the)g(\014rst)f(t)o(w)o(o)g(emphasize)f (that)h(the)h(corrections)g(in)f(the)g(pixel)970 1640 y(regularization)c(image)f(m)o(ust)h(b)q(e)i(small.)1032 1689 y(The)f(role)f(of)g(the)g(pixel)g(regularization)f(image)f(in)i (the)h(c)o(hain)970 1739 y(of)h(algorithms)f(is)i(sho)o(wn)f(in)h(Fig.) 11 b(1.)k(It)h(pro)o(vides)g(a)f(correction)970 1789 y(\()p Fo(\016)1004 1795 y Fi(i)1019 1789 y Fo(;)7 b(\016)1056 1795 y Fi(j)1073 1789 y Fw(\))12 b(to)g(b)q(e)h(applied)e(to)h(the)h (image)d(pixel)h(co)q(ordinates)i(\()p Fo(i)1932 1774 y Fj(0)1944 1789 y Fo(;)7 b(j)1982 1774 y Fj(0)1993 1789 y Fw(\))970 1839 y(to)14 b(con)o(v)o(ert)g(them)g(to)f(ideal)g(pixel)h (co)q(ordinates)g(\()p Fo(i;)7 b(j)r Fw(\).)14 b(That)g(is)970 1920 y(\()p Fo(i;)7 b(j)r Fw(\))12 b(=)g(\()p Fo(i)1140 1903 y Fj(0)1152 1920 y Fo(;)7 b(j)1190 1903 y Fj(0)1202 1920 y Fw(\))i(+)h(\()p Fo(\016)1303 1926 y Fi(i)1317 1920 y Fo(;)d(\016)1354 1926 y Fi(j)1371 1920 y Fw(\))538 b(\(A1\))970 2001 y(As)22 b(stated)g(in)e(Sect.)13 b(2.1,)20 b(in)o(tegral)g(pixel)h(co)q(ordinates)h(corre-)970 2051 y(sp)q(ond)14 b(to)g(the)h(cen)o(ter)g(of)e(the)i(pixel.)1032 2101 y(A)o(t)i(one)g(extreme,)g(the)g(pixel)f(regularization)g(image)f (could)970 2151 y(simply)g(store)j(a)e(correction)i(for)f(eac)o(h)g (pixel)f(in)h(the)g(image.)d(In)970 2201 y(practice)i(though,)f (su\016cien)o(t)g(accuracy)h(should)f(b)q(e)g(obtainable)970 2250 y(b)o(y)f(storing)f(corrections)i(for)e(a)g(regularly)h(spaced)g (sub-grid)g(and)970 2300 y(using)j(bilinear)f(in)o(terp)q(olation.)f(F) m(or)i(example,)e(if)h(a)g(correction)970 2356 y(w)o(ere)h(pro)o(vided) e(for)g(ev)o(ery)i(10)1459 2341 y(th)1515 2356 y(image)c(pixel)i(in)h (either)g(direc-)970 2406 y(tion)c(then)h(the)h(pixel)e(regularization) g(image)e(w)o(ould)i(ha)o(v)o(e)g(a)h(size)970 2456 y(only)g(2)h(p)q (er)h(cen)o(t)f(that)g(of)g(the)g(image.)1032 2506 y(The)h(pixel)f (regularization)g(image)f(stores)i(v)n(alues)g(on)f(a)g(reg-)970 2556 y(ular)26 b Fo(N)1101 2562 y Fi(i)1141 2556 y Fw(b)o(y)f Fo(N)1243 2562 y Fi(j)1287 2556 y Fw(grid)g(with)h(a)g(spacing)f(of)h (\()p Fo(s)1792 2562 y Fi(i)1806 2556 y Fo(;)7 b(s)1844 2562 y Fi(j)1861 2556 y Fw(\))26 b(image)970 2605 y(pixels)21 b(b)q(et)o(w)o(een)h(grid)e(elemen)o(ts)g(in)g(either)i(dimension.)c (Note)970 2655 y(that)g Fo(s)1083 2661 y Fi(i)1116 2655 y Fw(and)g Fo(s)1220 2661 y Fi(j)1256 2655 y Fw(are)h(not)f(required)h (to)f(b)q(e)h(in)o(tegers.)g(Supp)q(ose)p eop %%Page: 31 31 31 30 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(31)-116 1303 y Fm(Fig.)23 b(33.)f Fe(COD)d Fs(pro)r(jection)k (plotted)g(with)e(lines)i(in)f Fk(\013;)6 b(\016)22 b Fs(with)g(parameters)g(\(b)q(ottom)f(left:)g(A\))g Fk(\013)1433 1307 y Fd(P)1482 1303 y Fs(=)k(0,)20 b Fk(\016)1604 1307 y Fd(P)1654 1303 y Fs(=)k(30)1746 1287 y Ff(\016)1765 1303 y Fs(,)c Fk(\036)1819 1307 y Fd(P)1869 1303 y Fs(=)k(180)1980 1287 y Ff(\016)1999 1303 y Fs(,)-116 1348 y(\()p Fk(\013)-76 1352 y Ft(0)-44 1348 y Fs(=)16 b(0)p Fk(;)6 b(\016)55 1352 y Ft(0)88 1348 y Fs(=)16 b Fg(\000)p Fs(60)202 1333 y Ff(\016)220 1348 y Fs(\);)g(\(b)q(ottom)g(righ)o(t:B\))h Fk(\013)589 1352 y Fd(P)630 1348 y Fs(=)f(0,)f Fk(\016)738 1352 y Fd(P)780 1348 y Fs(=)g(30)863 1333 y Ff(\016)882 1348 y Fs(,)g Fk(\036)931 1352 y Fd(P)973 1348 y Fs(=)g(150)1075 1333 y Ff(\016)1094 1348 y Fs(,)g(\()p Fk(\013)1160 1352 y Ft(0)1193 1348 y Fs(=)h(49)p Fk(:)p Fs(1)1307 1333 y Ff(\016)1325 1348 y Fs(,)g Fk(\016)1369 1352 y Ft(0)1402 1348 y Fs(=)f Fg(\000)p Fs(48)p Fk(:)p Fs(6)1545 1333 y Ff(\016)1564 1348 y Fs(\);)g(\(top)h(left:)g(C\))g Fk(\013)1856 1352 y Fd(P)1897 1348 y Fs(=)f(45)1980 1333 y Ff(\016)1999 1348 y Fs(,)-116 1394 y Fk(\016)-99 1398 y Fd(P)-63 1394 y Fs(=)10 b(30)15 1378 y Ff(\016)34 1394 y Fs(,)i Fk(\036)80 1398 y Fd(P)116 1394 y Fs(=)f(180)214 1378 y Ff(\016)232 1394 y Fs(,)i(\()p Fk(\013)296 1398 y Ft(0)323 1394 y Fs(=)e(45)402 1378 y Ff(\016)420 1394 y Fs(,)i Fk(\016)461 1398 y Ft(0)489 1394 y Fs(=)d Fg(\000)p Fs(60)597 1378 y Ff(\016)615 1394 y Fs(\);)j(\(top)g(righ)o(t:)h(D\))f Fk(\013)923 1398 y Fd(P)958 1394 y Fs(=)e(0,)i Fk(\016)1059 1398 y Fd(P)1095 1394 y Fs(=)d(30)1173 1378 y Ff(\016)1192 1394 y Fs(,)i Fk(\036)1238 1398 y Fd(P)1274 1394 y Fs(=)f(75)1353 1378 y Ff(\016)1371 1394 y Fs(,)i(\()p Fk(\013)1435 1398 y Ft(0)1462 1394 y Fs(=)e(97)p Fk(:)p Fs(6)1571 1378 y Ff(\016)1589 1394 y Fs(,)h Fk(\016)1629 1398 y Ft(0)1657 1394 y Fs(=)f(13)p Fk(:)p Fs(0)1766 1378 y Ff(\016)1784 1394 y Fs(\).)-116 1528 y Fw(that)20 b Fo(P)6 b Fw(1)20 b(:)h(\()p Fo(i)117 1513 y Fj(0)117 1538 y Fu(1)136 1528 y Fo(;)7 b(j)174 1513 y Fj(0)172 1538 y Fu(1)191 1528 y Fw(\),)19 b Fo(P)6 b Fw(2)21 b(:)g(\()p Fo(i)376 1513 y Fj(0)376 1538 y Fu(1)408 1528 y Fw(+)14 b Fo(s)473 1534 y Fi(i)487 1528 y Fo(;)7 b(j)525 1513 y Fj(0)523 1538 y Fu(1)541 1528 y Fw(\),)20 b Fo(P)6 b Fw(3)20 b(:)h(\()p Fo(i)726 1513 y Fj(0)726 1538 y Fu(1)745 1528 y Fo(;)7 b(j)783 1513 y Fj(0)781 1538 y Fu(1)813 1528 y Fw(+)13 b Fo(s)877 1534 y Fi(j)895 1528 y Fw(\),)-116 1578 y(and)20 b Fo(P)6 b Fw(4)22 b(:)g(\()p Fo(i)111 1563 y Fj(0)111 1588 y Fu(1)143 1578 y Fw(+)14 b Fo(s)208 1584 y Fi(i)223 1578 y Fo(;)7 b(j)261 1563 y Fj(0)259 1588 y Fu(1)290 1578 y Fw(+)14 b Fo(s)355 1584 y Fi(j)373 1578 y Fw(\))21 b(are)g(the)g(image)d(pixel)i(co)q(ordi-)-116 1628 y(nates)g(of)f(the)h (four)f(grid)g(elemen)o(ts)h(surrounding)g(\()p Fo(i)754 1613 y Fj(0)766 1628 y Fo(;)7 b(j)804 1613 y Fj(0)815 1628 y Fw(\).)19 b(Let)-116 1677 y(\()p Fo(\016)-82 1683 y Fi(i)p Fu(1)-51 1677 y Fo(;)7 b(\016)-14 1683 y Fi(j)r Fu(1)19 1677 y Fw(\))p Fo(;)g(:)g(:)g(:)e Fw(\()p Fo(\016)143 1683 y Fi(i)p Fu(4)174 1677 y Fo(;)i(\016)211 1683 y Fi(j)r Fu(4)245 1677 y Fw(\))14 b(b)q(e)h(the)f(corresp)q(onding)h (o\013sets.)g(Then)-91 1749 y Fo(\016)-73 1755 y Fi(i)-38 1749 y Fw(=)21 b Fo(W)54 1755 y Fu(1)73 1749 y Fo(\016)91 1755 y Fi(i)p Fu(1)131 1749 y Fw(+)10 b Fo(W)212 1755 y Fu(2)230 1749 y Fo(\016)248 1755 y Fi(i)p Fu(2)288 1749 y Fw(+)g Fo(W)369 1755 y Fu(3)388 1749 y Fo(\016)406 1755 y Fi(i)p Fu(3)446 1749 y Fw(+)f Fo(W)526 1755 y Fu(4)545 1749 y Fo(\016)563 1755 y Fi(i)p Fu(4)839 1749 y Fw(\(A2\))-95 1811 y Fo(\016)-77 1817 y Fi(j)-38 1811 y Fw(=)21 b Fo(W)54 1817 y Fu(1)73 1811 y Fo(\016)91 1817 y Fi(j)r Fu(1)135 1811 y Fw(+)9 b Fo(W)215 1817 y Fu(2)234 1811 y Fo(\016)252 1817 y Fi(j)r Fu(2)296 1811 y Fw(+)g Fo(W)376 1817 y Fu(3)395 1811 y Fo(\016)413 1817 y Fi(j)r Fu(3)457 1811 y Fw(+)g Fo(W)537 1817 y Fu(4)556 1811 y Fo(\016)574 1817 y Fi(j)r Fu(4)839 1811 y Fw(\(A3\))-116 1874 y(where)-107 1923 y(\001)-72 1906 y Fj(0)-72 1934 y Fi(i)-38 1923 y Fw(=)21 b(\()p Fo(i)45 1906 y Fj(0)67 1923 y Fn(\000)9 b Fo(i)122 1906 y Fj(0)122 1934 y Fu(1)141 1923 y Fw(\))p Fo(=s)197 1929 y Fi(i)839 1923 y Fw(\(A4\))-111 1986 y(\001)-76 1969 y Fj(0)-76 1996 y Fi(j)-38 1986 y Fw(=)21 b(\()p Fo(j)50 1969 y Fj(0)72 1986 y Fn(\000)9 b Fo(j)132 1969 y Fj(0)130 1996 y Fu(1)149 1986 y Fw(\))p Fo(=s)205 1992 y Fi(j)839 1986 y Fw(\(A5\))-116 2048 y Fo(W)-77 2054 y Fu(1)-38 2048 y Fw(=)21 b(\(1)9 b Fn(\000)h Fw(\001)138 2031 y Fj(0)138 2058 y Fi(i)151 2048 y Fw(\))g Fn(\003)f Fw(\(1)g Fn(\000)g Fw(\001)329 2031 y Fj(0)329 2058 y Fi(j)346 2048 y Fw(\))477 b(\(A6\))-116 2110 y Fo(W)-77 2116 y Fu(2)-38 2110 y Fw(=)21 b(\001)50 2093 y Fj(0)50 2121 y Fi(i)73 2110 y Fn(\003)9 b Fw(\(1)g Fn(\000)g Fw(\001)225 2093 y Fj(0)225 2121 y Fi(j)243 2110 y Fw(\))580 b(\(A7\))-116 2173 y Fo(W)-77 2179 y Fu(3)-38 2173 y Fw(=)21 b(\(1)9 b Fn(\000)h Fw(\001)138 2155 y Fj(0)138 2183 y Fi(i)151 2173 y Fw(\))g Fn(\003)f Fw(\001)242 2155 y Fj(0)242 2183 y Fi(j)839 2173 y Fw(\(A8\))-116 2235 y Fo(W)-77 2241 y Fu(4)-38 2235 y Fw(=)21 b(\001)50 2218 y Fj(0)50 2245 y Fi(i)73 2235 y Fn(\003)9 b Fw(\001)138 2218 y Fj(0)138 2245 y Fi(j)839 2235 y Fw(\(A9\))-54 2307 y(F)m(or)i(the)i(in)o(v)o(erse)f (problem,)e(where)j(\()p Fo(i;)7 b(j)r Fw(\))12 b(is)g(kno)o(wn)f(and)h (it)f(is)-116 2356 y(desired)h(to)f(compute)g(\()p Fo(i)266 2341 y Fj(0)278 2356 y Fo(;)c(j)316 2341 y Fj(0)327 2356 y Fw(\),)k(in)o(v)o(ersion)g(of)f(the)i(ab)q(o)o(v)o(e)f(form)o(ulae) -116 2406 y(ma)o(y)h(b)q(e)i(done)g(b)o(y)g(iteration.)-54 2456 y(If)k(a)f(pixel)h(regularization)g(image)e(is)i(pro)o(vided,)f (it)h(should)-116 2506 y(co)o(v)o(er)k(the)h(whole)g(of)e(the)i(image)e (region;)h(no)g(extrap)q(olation)-116 2556 y(metho)q(ds)13 b(are)h(de\014ned)h(b)o(y)f(this)g(standard.)g(FITS-reading)f(soft-) -116 2605 y(w)o(are)d(pac)o(k)n(ages)h(will)d(pro)o(vide)j(implem)o(en) o(tation-sp)q(eci\014c)e(b)q(eha)o(v-)-116 2655 y(ior)15 b(in)g(the)i(ev)o(en)o(t)g(that)e(a)h(co)q(ordinate)g(calculation)f(is) h(required)970 1528 y(in)c(a)f(part)h(of)f(an)h(image)e(not)h(co)o(v)o (ered)i(b)o(y)f(a)f(pixel)g(regularization)970 1578 y(image.)e(This)i (migh)o(t)f(consist)i(of)e(simply)g(returning)h(a)g(fatal)g(error)970 1628 y(or)16 b(of)f(assuming)g(a)g(zero)i(correction;)f(the)h(c)o (hoice)f(ma)o(y)e(dep)q(end)970 1677 y(on)19 b(the)g(p)q(ositional)f (accuracy)i(required)f(with)g(resp)q(ect)i(to)e(the)970 1727 y(RMS)f(error)i(indicated)e(b)o(y)h(the)g(extan)o(t)g(p)q(ortion)f (of)g(the)h(pixel)970 1777 y(regularization)13 b(image.)1032 1843 y(The)g(pixel)f(regularization)g(image)f(will)g(b)q(e)i(implemen)o (ted)d(as)970 1892 y(a)20 b(FITS)h(image)d(extension)k(\(P)o(onz)f Fr(et)f(al.)p Fw(,)g(1994)o(\).)g(The)h(pixel)970 1942 y(regularization)15 b(image)e(will)h(con)o(tain)h(t)o(w)o(o)g(image)e (planes)j(of)e Fo(N)1995 1948 y Fi(i)970 1992 y Fw(columns)j(and)h Fo(N)1254 1998 y Fi(j)1290 1992 y Fw(ro)o(ws)g(eac)o(h.)h(The)f (\014rst)h(plane)f(will)f(con)o(tain)970 2042 y(the)d Fo(\016)1059 2048 y Fi(i)1087 2042 y Fw(and)f(the)h(second)g(the)g Fo(\016)1460 2048 y Fi(j)1478 2042 y Fw(.)f(The)h(storage)g(order)g (will)d(b)q(e)j(as)970 2092 y(for)f(FITS)g(images)f(so)h(that)h(the)f (\014rst)h(ro)o(w)g(of)e(the)i(pixel)f(regular-)970 2141 y(ization)j(image)f(corresp)q(onds)k(\(lo)q(osely)e(sp)q(eaking\))g(to) g(the)h(\014rst)970 2191 y(ro)o(w)c(of)f(the)i(image.)1032 2257 y(The)k(pixel)e(regularization)g(image)f(will)g(con)o(tain)h (su\016cien)o(t)970 2307 y(information)c(to)i(asso)q(ciate)h (correction)g(grid)f(co)q(ordinates)h(with)970 2356 y(image)k(pixel)i (co)q(ordinates.)g(This)g(will)f(b)q(e)i(implem)o(en)o(ted)d(via)970 2406 y Fp(CRPIX)p Fr(n)p Fw(,)c Fp(CRVAL)p Fr(n)p Fw(,)g(and)i Fp(CDELT)p Fr(n)e Fw(k)o(eyw)o(ords)i(in)f(the)h(pixel)e(reg-)970 2456 y(ularization)i(image)f(header.)j(The)f Fp(CRPIX)p Fr(n)g Fw(de\014ne)h(a)e(lo)q(cation)970 2506 y(\()p Fo(i)1000 2491 y Fj(00)1000 2516 y Fu(0)1022 2506 y Fo(;)7 b(j)1060 2491 y Fj(00)1058 2516 y Fu(0)1081 2506 y Fw(\))18 b(on)g(the)h(correction)h(grid)d(whic)o(h)i(corresp)q(onds)h(to)e(the) 970 2556 y(image)10 b(pixel)i(co)q(ordinate)g(\()p Fo(i)1419 2541 y Fj(0)1419 2566 y Fu(0)1438 2556 y Fo(;)7 b(j)1476 2541 y Fj(0)1474 2566 y Fu(0)1492 2556 y Fw(\))12 b(de\014ned)h(b)o(y)f (the)h Fp(CRVAL)p Fr(n)p Fw(,)e(and)970 2605 y(the)j Fp(CDELT)p Fr(n)e Fw(sp)q(ecify)i(the)f(grid)g(compression)f(ratio,)g (\()p Fo(s)1855 2611 y Fi(i)1870 2605 y Fo(;)7 b(s)1908 2611 y Fi(j)1925 2605 y Fw(\))13 b(de-)970 2655 y(\014ned)i(ab)q(o)o(v) o(e.)f(The)h(relationship)f(b)q(et)o(w)o(een)i(correction)f(grid)g(co-) p eop %%Page: 32 32 32 31 bop -116 -192 a Fs(32)430 b(Greisen)14 b(&)e(Calabretta:)i (Represen)o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 1003 y Fm(Fig.)h(34.)g Fe(AIT)e Fs(pro)r(jection)j(plotted)f(with)g (lines)h(in)f Fk(\013;)6 b(\016)16 b Fs(with)f(parameters)g(\(b)q (ottom)f(left:)h(A\))f Fk(\013)1337 1007 y Ft(0)1366 1003 y Fs(=)f(0,)h Fk(\016)1470 1007 y Ft(0)1500 1003 y Fs(=)f Fg(\000)p Fs(60)1611 987 y Ff(\016)1630 1003 y Fs(,)g Fk(\036)1677 1007 y Fd(P)1716 1003 y Fs(=)f(180)1815 987 y Ff(\016)1834 1003 y Fs(,)i(\()p Fk(\013)1899 1007 y Fd(P)1937 1003 y Fs(=)f(0,)-116 1048 y Fk(\016)-99 1052 y Fd(P)-60 1048 y Fs(=)h(30)22 1033 y Ff(\016)41 1048 y Fs(\);)g(\(b)q(ottom)i(righ)o(t:)f(B\))g Fk(\013)421 1052 y Ft(0)452 1048 y Fs(=)f(49)p Fk(:)p Fs(1)564 1033 y Ff(\016)583 1048 y Fs(,)g Fk(\016)625 1052 y Ft(0)657 1048 y Fs(=)g Fg(\000)p Fs(48)p Fk(:)p Fs(6)799 1033 y Ff(\016)817 1048 y Fs(,)g Fk(\036)865 1052 y Fd(P)905 1048 y Fs(=)g(150)1006 1033 y Ff(\016)1024 1048 y Fs(,)h(\()p Fk(\013)1090 1052 y Fd(P)1129 1048 y Fs(=)f(0,)h Fk(\016)1235 1052 y Fd(P)1275 1048 y Fs(=)f(30)1357 1033 y Ff(\016)1375 1048 y Fs(\);)g(\(top)i(left:)f(C\))f Fk(\013)1663 1052 y Ft(0)1694 1048 y Fs(=)g(45)1776 1033 y Ff(\016)1795 1048 y Fs(,)g Fk(\016)1837 1052 y Ft(0)1869 1048 y Fs(=)g Fg(\000)p Fs(60)1981 1033 y Ff(\016)1999 1048 y Fs(,)-116 1094 y Fk(\036)-93 1098 y Fd(P)-57 1094 y Fs(=)c(180)40 1078 y Ff(\016)59 1094 y Fs(,)i(\()p Fk(\013)122 1098 y Fd(P)158 1094 y Fs(=)e(45)236 1078 y Ff(\016)255 1094 y Fs(,)i Fk(\016)295 1098 y Fd(P)332 1094 y Fs(=)e(30)410 1078 y Ff(\016)428 1094 y Fs(\);)j(\(top)g(righ)o(t:)h(D\))f Fk(\013)736 1098 y Ft(0)763 1094 y Fs(=)e(97)p Fk(:)p Fs(6)872 1078 y Ff(\016)890 1094 y Fs(,)h Fk(\016)930 1098 y Ft(0)958 1094 y Fs(=)f(13)p Fk(:)p Fs(0)1067 1078 y Ff(\016)1085 1094 y Fs(,)i Fk(\036)1132 1098 y Fd(P)1168 1094 y Fs(=)d(75)1246 1078 y Ff(\016)1265 1094 y Fs(,)i(\()p Fk(\013)1328 1098 y Fd(P)1364 1094 y Fs(=)e(0,)j Fk(\016)1464 1098 y Fd(P)1500 1094 y Fs(=)e(30)1579 1078 y Ff(\016)1597 1094 y Fs(\).)-116 1228 y Fw(ordinates)j(and)g(image)d(pixel)j(co)q (ordinates)g(is)g(therefore)-111 1306 y Fo(i)-97 1289 y Fj(0)-65 1306 y Fw(=)21 b Fo(s)7 1312 y Fi(i)31 1306 y Fn(\003)9 b Fw(\()p Fo(i)91 1289 y Fj(00)122 1306 y Fn(\000)g Fo(i)177 1289 y Fj(00)177 1316 y Fu(0)199 1306 y Fw(\))g(+)h Fo(i)280 1289 y Fj(0)280 1316 y Fu(0)818 1306 y Fw(\(A10\))-116 1368 y Fo(j)-97 1351 y Fj(0)-65 1368 y Fw(=)21 b Fo(s)7 1374 y Fi(j)35 1368 y Fn(\003)9 b Fw(\()p Fo(j)100 1351 y Fj(00)131 1368 y Fn(\000)g Fo(j)191 1351 y Fj(00)189 1378 y Fu(0)213 1368 y Fw(\))g(+)h Fo(j)299 1351 y Fj(0)297 1378 y Fu(0)818 1368 y Fw(\(A11\))-54 1446 y(A)i(pixel)g(regularization)g(image)f(ma)o(y)f(b)q(e)k(asso)q (ciated)f(with)f(a)-116 1496 y(sub-image)h(extracted)k(from)c(an)i (image)e(b)o(y)h(suitably)h(c)o(hanging)-116 1546 y(the)d Fp(CRPIX)p Fr(n)p Fw(,)e Fp(CRVAL)p Fr(n)p Fw(,)g(and)h Fp(CDELT)p Fr(n)p Fw(.)f(If)h(size)h(is)g(imp)q(ortan)o(t,)d(this)-116 1595 y(could)14 b(also)g(b)q(e)i(accompanied)d(b)o(y)i(a)f(reduction)i (in)e(the)i(n)o(um)o(b)q(er)-116 1645 y(of)d(columns)g(and)g(ro)o(ws.) -54 1695 y(The)f(header)h(sho)o(wn)f(in)f(T)m(able)h(7)f(describ)q(es)j (a)e(pixel)f(regular-)-116 1745 y(ization)k(image)g(with)h(image)f(v)n (alues)h(giv)o(en)g(in)g(terms)g(of)g(single)-116 1795 y(precision)d(\(4-b)o(yte\))g(\015oating-p)q(oin)o(t)f(n)o(um)o(b)q (ers.)g(Angular)h(brac)o(k-)-116 1844 y(ets)20 b(denote)g(form)o(ulae)d (expressed)k(in)e(terms)g(of)g(the)h(v)n(ariables)-116 1894 y(de\014ned)g(ab)q(o)o(v)o(e)g(for)f(quan)o(tities)h(whic)o(h)f(m) o(ust)g(b)q(e)i(supplied)e(in)-116 1944 y(n)o(umerical)12 b(form.)-116 2054 y Fx(B.)k(Alternate)26 b(FITS)j(image)e(represen)o (tatio)o(ns:)f(pixel)-53 2104 y(list)14 b(and)h(v)o(ector)g(column)f (elemen)o(ts)615 2089 y Fa(4)-116 2181 y Fw(In)20 b(addition)f(to)h (the)g(image)f(format)f(discussed)k(in)d(the)i(main)-116 2230 y(b)q(o)q(dy)11 b(of)f(this)i(pap)q(er)f(\()p Fr(i.e.)p Fw(,)g(an)f(N-dimensional)f(arra)o(y)i(in)g(a)g(FITS)-116 2280 y(primary)18 b(arra)o(y)i(or)g(FITS)g Fp(IMAGE)f Fw(extension\))i(there)h(are)e(t)o(w)o(o)-116 2330 y(other)10 b(FITS)h(image)d(represen)o(tations)k(that)e(are)h(used)g(commonly)-116 2380 y(b)o(y)i(the)i(astronomical)d(comm)o(uni)o(t)o(y:)-103 2441 y(1.)k(a)k(m)o(ulti{dim)o(ensional)d(v)o(ector)k(in)f(a)g(single)g (elemen)o(t)g(of)g(a)-54 2491 y(FITS)14 b(binary)f(table,)h(and)p -116 2520 237 2 v -116 2548 a Ft(4)-59 2564 y Fs(Con)o(tributed)k(b)o (y)f(William)j(P)o(ence,)c(Arnold)i(Rots,)e(and)h(Lorella)-116 2610 y(Angelini)h(of)e(the)g(NASA)f(Go)q(ddard)j(Space)e(Fligh)o(t)i (Cen)o(ter,)d(Green-)-116 2655 y(b)q(elt,)e(MD)g(20771.)984 1228 y Fw(2.)i(a)g(tabulated)g(list)g(of)f(pixel)h(co)q(ordinates)h(in) e(a)h(FITS)g(ASCI)q(I)1032 1278 y(or)f(binary)g(table.)1032 1342 y(The)20 b(purp)q(ose)g(of)f(this)g(app)q(endix)g(is)g(to)g (de\014ne)h(a)f(naming)970 1392 y(con)o(v)o(en)o(tion)12 b(for)g(the)h(co)q(ordinate)f(system)g(k)o(eyw)o(ords)h(to)f(b)q(e)h (used)970 1441 y(with)h(these)h(alternate)f(image)e(formats.)970 1548 y Fr(B.1.)j(Multi-dimensional)g(ve)n(ctor)f(in)h(a)g(binary)g (table)970 1625 y Fw(A)h(v)o(ector)h(column)d(in)i(a)g(binary)f(table)h (\()p Fp(BINTABLE)p Fw(\))f(extension)970 1675 y(can)k(b)q(e)f(used)i (to)e(store)h(a)f(m)o(ulti-dim)n(ensional)d(image)h(in)i(eac)o(h)970 1725 y(elemen)o(t)13 b(\()p Fr(i.e.)p Fw(,)f(eac)o(h)i(ro)o(w\))g(of)f (the)h(column.)d(In)j(the)g(simple)e(case)970 1774 y(in)e(whic)o(h)h (all)e(the)i(images)e(ha)o(v)o(e)h(the)h(same)f(\014xed)h(size,)g(the)g Fp(TDIM)p Fr(n)970 1824 y Fw(k)o(eyw)o(ord)20 b(can)g(b)q(e)h(used)g (to)f(sp)q(ecify)g(the)h(dimensions.)d(In)i(the)970 1874 y(more)c(general)g(case,)h(a)g(v)n(ariable)e(length)i(v)o(ector)g(ma)o (y)d(b)q(e)k(used)970 1924 y(to)c(store)h(di\013eren)o(t-sized)g (images)e(within)g(the)h(same)f(column.)1032 1974 y(Because)21 b(t)o(w)o(o)d(or)h(more)f(columns)g(in)g(a)g(binary)h(table)f(can)970 2023 y(con)o(tain)f(images,)e(the)j(naming)d(con)o(v)o(en)o(tion)h(for) h(these)i(co)q(ordi-)970 2073 y(nate)g(system)g(k)o(eyw)o(ords)g(m)o (ust)f(enco)q(de)i(the)g(column)d(n)o(um)o(b)q(er)970 2123 y(con)o(taining)11 b(the)h(image)e(to)i(whic)o(h)g(the)g(k)o(eyw)o (ord)g(applies)f(as)h(w)o(ell)970 2173 y(as)g(the)g(axis)f(n)o(um)o(b)q (er)g(within)g(the)h(image.)d(The)j(naming)d(con)o(v)o(en-)970 2223 y(tion)k(describ)q(ed)i(here)f(uses)h(the)f(k)o(eyw)o(ord)f (pre\014x)h(to)f(sp)q(ecify)h(the)970 2272 y(axis)e(n)o(um)o(b)q(er)g (and)g(the)h(k)o(eyw)o(ord)g(su\016x)f(to)g(sp)q(ecify)h(the)g(column) 970 2322 y(n)o(um)o(b)q(er)h(con)o(taining)f(the)h(image)f(\()p Fr(e.g.)p Fw(,)g(the)i Fp(2CDLT15)e Fw(k)o(eyw)o(ord)970 2372 y(applies)k(to)h(the)g(second)g(axis)f(of)g(the)h(image)d(in)i (column)f(15)h(of)970 2422 y(the)e(table\).)970 2529 y Fr(B.2.)g(T)m(abulate)n(d)g(list)f(of)g(pixels)970 2605 y Fw(An)e(image)e(ma)o(y)f(also)i(b)q(e)h(represen)o(ted)j(as)c(a) h(list)f(of)g Fo(i)h Fw(and)f Fo(j)j Fw(pixel)970 2655 y(co)q(ordinates)c(in)f(a)g(binary)g(or)h(ASCI)q(I)g(table)f (extension.)h(This)f(rep-)p eop %%Page: 33 33 33 32 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(33)-116 2454 y Fm(Fig.)15 b(35.)g Fe(CSC)e Fs(pro)r(jection)j (plotted)f(with)g(lines)h(in)f Fk(\013;)6 b(\016)16 b Fs(with)f(parameters)g(\(b)q(ottom)f(left:)h(A\))f Fk(\013)1337 2458 y Ft(0)1366 2454 y Fs(=)f(0,)h Fk(\016)1470 2458 y Ft(0)1500 2454 y Fs(=)f Fg(\000)p Fs(60)1611 2438 y Ff(\016)1630 2454 y Fs(,)g Fk(\036)1677 2458 y Fd(P)1716 2454 y Fs(=)f(180)1815 2438 y Ff(\016)1834 2454 y Fs(,)i(\()p Fk(\013)1899 2458 y Fd(P)1937 2454 y Fs(=)f(0,)-116 2499 y Fk(\016)-99 2503 y Fd(P)-60 2499 y Fs(=)h(30)22 2483 y Ff(\016)41 2499 y Fs(\);)g(\(b)q(ottom)i(righ)o(t:)f(B\))g Fk(\013)421 2503 y Ft(0)452 2499 y Fs(=)f(49)p Fk(:)p Fs(1)564 2483 y Ff(\016)583 2499 y Fs(,)g Fk(\016)625 2503 y Ft(0)657 2499 y Fs(=)g Fg(\000)p Fs(48)p Fk(:)p Fs(6)799 2483 y Ff(\016)817 2499 y Fs(,)g Fk(\036)865 2503 y Fd(P)905 2499 y Fs(=)g(150)1006 2483 y Ff(\016)1024 2499 y Fs(,)h(\()p Fk(\013)1090 2503 y Fd(P)1129 2499 y Fs(=)f(0,)h Fk(\016)1235 2503 y Fd(P)1275 2499 y Fs(=)f(30)1357 2483 y Ff(\016)1375 2499 y Fs(\);)g(\(top)i(left:)f(C\))f Fk(\013)1663 2503 y Ft(0)1694 2499 y Fs(=)g(45)1776 2483 y Ff(\016)1795 2499 y Fs(,)g Fk(\016)1837 2503 y Ft(0)1869 2499 y Fs(=)g Fg(\000)p Fs(60)1981 2483 y Ff(\016)1999 2499 y Fs(,)-116 2545 y Fk(\036)-93 2549 y Fd(P)-57 2545 y Fs(=)c(180)40 2529 y Ff(\016)59 2545 y Fs(,)i(\()p Fk(\013)122 2549 y Fd(P)158 2545 y Fs(=)e(45)236 2529 y Ff(\016)255 2545 y Fs(,)i Fk(\016)295 2549 y Fd(P)332 2545 y Fs(=)e(30)410 2529 y Ff(\016)428 2545 y Fs(\);)j(\(top)g(righ)o (t:)h(D\))f Fk(\013)736 2549 y Ft(0)763 2545 y Fs(=)e(97)p Fk(:)p Fs(6)872 2529 y Ff(\016)890 2545 y Fs(,)h Fk(\016)930 2549 y Ft(0)958 2545 y Fs(=)f(13)p Fk(:)p Fs(0)1067 2529 y Ff(\016)1085 2545 y Fs(,)i Fk(\036)1132 2549 y Fd(P)1168 2545 y Fs(=)d(75)1246 2529 y Ff(\016)1265 2545 y Fs(,)i(\()p Fk(\013)1328 2549 y Fd(P)1364 2545 y Fs(=)e(0,)j Fk(\016)1464 2549 y Fd(P)1500 2545 y Fs(=)e(30)1579 2529 y Ff(\016)1597 2545 y Fs(\).)p eop %%Page: 34 34 34 33 bop -116 -192 a Fs(34)430 b(Greisen)14 b(&)e(Calabretta:)i (Represen)o(tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)-116 -67 y Fm(T)l(able)g(4.)f Fs(Co)q(ordinate)i(k)o(eyw)o(ords)-116 4 y Fw(Keyw)o(ord)p -116 19 162 2 v 63 w(Use)p 108 11 66 2 v 473 w(Status)p 646 11 116 2 v 130 w(Commen)o(ts)p 891 11 193 2 v -116 56 a Fp(CRVAL)p Fr(i)101 b Fw(v)n(alue)13 b(at)h(reference)i(p)q(oin)o(t)111 b(clari\014ed)102 b(meaning)13 b(of)g(reference)j(p)q(oin)o(t)e(forced)g(b)o(y)g(pro)r (jection;)g(no)f(default.)-116 105 y Fp(CRPIX)p Fr(i)101 b Fw(pixel)13 b(of)g(reference)k(p)q(oin)o(t)121 b(clari\014ed)102 b(meaning)13 b(of)g(reference)j(p)q(oin)o(t)e(forced)g(b)o(y)g(pro)r (jection;)g(no)f(default.)-116 155 y Fp(CDELT)p Fr(i)101 b Fw(incremen)o(t)13 b(at)h(ref.)f(p)q(oin)o(t)128 b(clari\014ed)102 b(meaning)13 b(of)g(reference)j(p)q(oin)o(t)e(forced)g(b)o(y)g(pro)r (jection;)g(no)f(default.)-116 205 y Fp(CROTA)p Fr(i)101 b Fw(rotation)13 b(at)h(ref.)e(p)q(oin)o(t)162 b(deprecated)51 b(replaced)15 b(b)o(y)f Fp(PC)p Fr(iiijjj)p Fw(.)-116 255 y Fp(CTYPE)p Fr(i)101 b Fw(co)q(ordinate/pro)r(jection)14 b(t)o(yp)q(e)51 b(clari\014ed)102 b(for)14 b(spherical)g(co)q (ordinates,)g(\014rst)h(4)f(c)o(haracters)h(giv)o(e)f(\\standard)891 305 y(system")g(used)h(in)e Fp(CRVAL)p Fr(n)p Fw(,)g(second)i(4)e(c)o (haracters)j(giv)o(e)d(t)o(yp)q(e)i(of)891 354 y(pro)r(jection)g(as)f (in)f(T)m(able)f(5;)h(no)h(default.)-116 404 y Fp(CUNIT)p Fr(i)101 b Fw(units)14 b(of)f(co)q(ordinate)h(v)n(alues)73 b(new)174 b(c)o(haracter-v)n(alued;)15 b(k)o(eep)f(it)g(simple)e (please;)891 454 y(ignored)i(for)g(angles)f(whic)o(h)h(are)h(alw)o(a)o (ys)d(degrees)-116 504 y Fp(PC)p Fr(iiijjj)102 b Fw(co)q(ordinate)14 b(incremen)o(t)155 b(new)174 b(con)o(v)o(erts)15 b(pixel)f(n)o(um)o(b)q (er)f(to)h(pixels)f(along)g(true)i(co)q(ordinates;)891 554 y(default)f(=)e(0\()p Fo(iii)g Fn(6)p Fw(=)g Fo(j)r(j)r(j)r Fw(\))p Fo(;)7 b 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Fp(ELT)p Fr(i)86 b Fw(incremen)o(t)13 b(at)h(ref.)f(p)q(oin)o(t)128 b(new)174 b(secondary)15 b(co)q(ordinate)g(description;)f(no)f(default.)-116 1849 y Fp(C)p Fr(m)s Fp(YPE)p Fr(i)86 b Fw(co)q(ordinate/pro)r(jection) 14 b(t)o(yp)q(e)51 b(new)174 b(secondary)15 b(co)q(ordinate)g (description;)f(no)f(default.)-116 1899 y Fp(C)p Fr(m)s Fp(NIT)p Fr(i)86 b Fw(units)14 b(of)f(co)q(ordinate)h(v)n(alues)73 b(new)174 b(secondary)15 b(co)q(ordinate)g(description;)891 1948 y(no)f(default)g(except)h(angles)f(are)g(in)f(degrees.)-116 2088 y(resen)o(tation)h(is)g(frequen)o(tly)g(used)h(in)e(high-energy)h (astroph)o(ysics)-116 2138 y(as)f(a)g(w)o(a)o(y)f(of)g(recording)i(the) g(p)q(osition)e(and)h(other)h(prop)q(erties)g(of)-116 2188 y(individually)8 b(detected)14 b(photons.)d(This)g(image)e(format) h(requires)-116 2237 y(a)16 b(minim)n(um)c(of)k(2)h(table)f(columns)g (whic)o(h)g(giv)o(e)h(the)g Fo(i)g Fw(\(axis)f(1\))-116 2287 y(and)c Fo(j)j Fw(\(axis)d(2\))h(pixel)f(co)q(ordinate)h(of)f(the) 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y(=)12 b Fo(j)r Fw(\).)1032 2307 y(A)18 b(v)n(ariation)e(on)h (this)h(pixel)f(list)g(format)f(ma)o(y)f(b)q(e)j(used)h(to)970 2356 y(explicitly)c(sp)q(ecify)h(the)h(in)o(tensit)o(y)e(v)n(alue)g(of) g(eac)o(h)i(image)d(pixel.)970 2406 y(This)d(case)g(requires)h(at)f (least)g(three)h(table)e(columns)g(whic)o(h)g(sp)q(ec-)970 2456 y(ify)16 b(the)h(axis)g(1)f(co)q(ordinate,)h(the)g(axis)f(2)h(co)q (ordinate,)f(and)h(the)970 2506 y(v)n(alue)f(of)g(the)h(pixel)f(at)g (that)h(co)q(ordinate.)f(In)h(this)g(represen)o(ta-)970 2556 y(tion)h(eac)o(h)g(pixel)g(co)q(ordinate)g(w)o(ould)g(only)f(b)q (e)i(listed)f(at)g(most)970 2605 y(once)c(in)f(the)g(table;)g(pixels)g (with)f(a)h(v)n(alue)f(=)g(0)h(ma)o(y)e(b)q(e)j(omitted)970 2655 y(en)o(tirely)g(from)e(the)j(table)f(to)f(conserv)o(e)j(space.)p eop %%Page: 35 35 35 34 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(35)-116 -67 y Fm(T)l(able)14 b(5.)f 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y(con)o(tains)c(a)h(300)f Fn(\002)h Fw(200)e(pixel)i(image)d(of)j (a)f(di\013eren)o(t)i(region)e(on)-116 1505 y(the)h(sky)m(.)g(This)g (migh)o(t)e(represen)o(t)k(a)e(set)h(of)f(smaller)e(images)h(ex-)-116 1555 y(tracted)k(from)d(a)h(single)g(larger)h(CCD)f(image.)f(In)i(this) f(case)i(all)-116 1605 y(the)13 b(co)q(ordinate)g(system)g(parameters)g (except)h(for)f(the)h(pixel)e(co-)-116 1655 y(ordinate)e(of)g(the)h (reference)i(p)q(oin)o(t)d(are)h(the)g(same)f(for)g(eac)o(h)h(image) -116 1705 y(and)k(are)h(giv)o(en)e(as)i(header)g(k)o(eyw)o(ords.)g(The) f(pixel)g(co)q(ordinates)-116 1754 y(of)f(the)i(reference)h(p)q(oin)o (t)e(are)g(di\013eren)o(t)h(for)f(eac)o(h)g(image,)e(there-)-116 1804 y(fore)h(they)g(are)h(giv)o(en)e(in)h(the)g Fp(1CRPX3)f Fw(and)h Fp(2CRPX3)f Fw(columns)f(of)-116 1854 y(the)i(table.)-54 1906 y(Finally)m(,)c(T)m(able)i(11)g(sho)o(ws)h(the)g(header)h(for)e (the)h(same)f(image)-116 1956 y(when)j(it)f(is)h(giv)o(en)f(in)h(the)g 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2454 y(minim)n(um)8 b(and)k(maxim)n(um)c (legal)j(v)n(alues)h(in)f(XPOS)i(and)f(YPOS)-116 2504 y(columns)h(and)h(are)g(useful)h(for)e(determining)g(the)i(range)g(of)e (eac)o(h)-116 2554 y(axis)g(when)h(constructing)h(the)g(image)d (histogram.)-54 2605 y(This)k(k)o(eyw)o(ord)g(naming)d(con)o(v)o(en)o (tion)j(is)g(optional)f(and)g(will)-116 2655 y(not)e(preclude)j(other)e (con)o(v)o(en)o(tions.)970 -93 y Fx(C.)j(Co)q(ordinate)d(represen)o (tati)o(on)e(com)o(bined)970 -15 y Fw(T)m(o)19 b(solv)o(e)g(the)h (general)g(case)g(problem)e(in)h(whic)o(h)g(an)o(y)g(t)o(w)o(o)g(of)970 35 y(the)13 b(four)f(pixel)f(and)h(celestial)g(co)q(ordinates)h(are)f (kno)o(wn)g(and)g(the)970 85 y(other)i(t)o(w)o(o)f(desired,)h(w)o(e)g (m)o(ust)e(com)o(bine)g(Eqs.)h(3)g(and)g(4)g(\(or)g(5)h(or)970 135 y(6\))d(with)f(the)h(equations)g(for)f(the)i(particular)e(pro)r (jection)h(relating)970 185 y(the)18 b(natural)f(longitude)g(and)g (latitude)g(\()p Fo(\036;)7 b(\022)q Fw(\))17 b(with)g(the)h(linear)970 234 y(displacemen)o(ts)10 b(\()p Fo(x;)d(y)q Fw(\).)i(In)h(all)f (equations)g(b)q(elo)o(w,)g(w)o(e)h(refer)h(to)f(the)970 284 y(longitude-lik)o(e)16 b(axis)g(with)h(sym)o(b)q(ols)e Fo(i;)7 b(x;)g(\013)16 b Fw(and)g(the)i(latitude-)970 334 y(lik)o(e)13 b(axis)h(with)f(sym)o(b)q(ols)f Fo(j;)7 b(y)q(;)g(\016)16 b Fw(for)d(pixel,)g(relativ)o(e)h(linear,)f(and)970 384 y(standard)h(spherical)h(co)q(ordinates,)f(resp)q(ectiv)o(ely)m(.) 1032 434 y(F)m(or)d(the)h(zenithal,)f(p)q(ersp)q(ectiv)o(e)j(pro)r (jection)e(\()p Fp(AZP)p Fw(\),)e(w)o(e)i(ha)o(v)o(e)970 527 y Fo(x)21 b Fw(=)g(\()p Fp(PC)p Fr(xxxxxx)q Fw(\()p Fo(i)10 b Fn(\000)f Fo(i)1338 533 y Fu(0)1357 527 y Fw(\))g(+)h Fp(PC)p Fr(xxxyyy)q Fw(\()p Fo(j)i Fn(\000)d Fo(j)1689 533 y Fu(0)1708 527 y Fw(\)\))p Fp(CDELT)p Fr(x)972 590 y Fo(y)23 b Fw(=)e(\()p Fp(PC)p Fr(yyyxxx)q Fw(\()p Fo(i)9 b Fn(\000)h Fo(i)1341 596 y Fu(0)1360 590 y Fw(\))f(+)h Fp(PC)p Fr(yyyyyy)p Fw(\()p Fo(j)i Fn(\000)e Fo(j)1695 596 y Fu(0)1713 590 y Fw(\)\))p Fp(CDELT)p 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Fu(0)1338 1250 y Fw(\))j(=)1012 1336 y Fn(\000)p Fw(\()p Fo(\026)d Fw(+)h(1\))1395 1308 y(cos)e Fo(\016)h Fw(sin\()p Fo(\013)g Fn(\000)g Fo(\013)1656 1314 y Fi(P)1684 1308 y Fw(\))p 1178 1327 740 2 v 1178 1365 a Fo(\026)g Fw(+)g(sin)e Fo(\016)i Fw(sin)e Fo(\016)1414 1371 y Fi(P)1452 1365 y Fw(+)i(cos)f Fo(\016)h Fw(cos)e Fo(\016)1663 1371 y Fi(P)1698 1365 y Fw(cos)q(\()p Fo(\013)i Fn(\000)g Fo(\013)1874 1371 y Fi(P)1901 1365 y Fw(\))970 1423 y Fo(C)s Fw(\()p Fo(i)g Fn(\000)h Fo(i)1098 1429 y Fu(0)1117 1423 y Fw(\))f(+)h Fo(D)q Fw(\()p Fo(j)i Fn(\000)e Fo(j)1323 1429 y Fu(0)1341 1423 y Fw(\))i(=)526 b(\(C2\))1012 1509 y Fn(\000)p Fw(\()p Fo(\026)9 b Fw(+)h(1\))1216 1481 y(cos)d Fo(\016)i Fw(sin)e Fo(\016)1381 1487 y Fi(P)1416 1481 y Fw(cos)q(\()p Fo(\013)i Fn(\000)g Fo(\013)1592 1487 y Fi(P)1619 1481 y Fw(\))h Fn(\000)f Fw(sin)e Fo(\016)i Fw(cos)f Fo(\016)1852 1487 y Fi(P)p 1178 1499 V 1178 1537 a Fo(\026)h Fw(+)g(sin)e Fo(\016)i Fw(sin)e Fo(\016)1414 1543 y Fi(P)1452 1537 y Fw(+)i(cos)f Fo(\016)h Fw(cos)e Fo(\016)1663 1543 y Fi(P)1698 1537 y Fw(cos)q(\()p Fo(\013)i Fn(\000)g Fo(\013)1874 1543 y Fi(P)1901 1537 y Fw(\))1929 1509 y Fo(;)970 1624 y Fw(where)975 1717 y Fo(A)20 b Fn(\021)1113 1689 y Fo(\031)p 1084 1707 82 2 v 1084 1745 a Fw(180)1147 1733 y Fj(\016)1171 1683 y Fl(\000)1190 1717 y Fw(cos)7 b Fo(\036)1277 1723 y Fi(P)1305 1717 y Fp(PC)p Fr(xxxxxx)h Fp(CDELT)p Fr(x)1266 1797 y Fw(+)h(sin)e Fo(\036)1390 1803 y Fi(P)1418 1797 y Fp(PC)p Fr(yyyxxx)g Fp(CDELT)p Fr(y)1716 1763 y Fl(\001)972 1874 y Fo(B)23 b Fn(\021)1113 1846 y Fo(\031)p 1084 1865 V 1084 1903 a Fw(180)1147 1891 y Fj(\016)1171 1841 y Fl(\000)1190 1874 y Fw(cos)7 b Fo(\036)1277 1880 y Fi(P)1305 1874 y Fp(PC)p Fr(xxxyyy)g Fp(CDELT)p Fr(x)1266 1954 y Fw(+)i(sin)e Fo(\036)1390 1960 y Fi(P)1418 1954 y Fp(PC)p Fr(yyyyyy)g Fp(CDELT)p Fr(y)1718 1921 y Fl(\001)973 2032 y Fo(C)23 b Fn(\021)1113 2003 y Fo(\031)p 1084 2022 V 1084 2060 a Fw(180)1147 2048 y Fj(\016)1171 1998 y Fl(\000)1190 2032 y 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(,)d(ho)o(w)o(ev)o(er.)h(F)m(or)g(the)970 2562 y(zenithal)14 b(equidistan)o(t)g(pro)r(jection)g(\()p Fp(ARC)p Fw(\),)f(Eqs.)g(C2)g (b)q(ecome)970 2655 y Fo(A)p Fw(\()p Fo(i)d Fn(\000)f Fo(i)1096 2661 y Fu(0)1115 2655 y Fw(\))h(+)f Fo(B)r Fw(\()p Fo(j)k Fn(\000)c Fo(j)1319 2661 y Fu(0)1338 2655 y Fw(\))j(=)p eop %%Page: 39 39 39 38 bop 352 -192 a Fs(Greisen)14 b(&)e(Calabretta:)i(Represen)o (tations)i(of)c(celestial)k(co)q(ordinates)f(in)f(FITS)429 b(39)-116 -67 y Fm(T)l(able)14 b(9.)f Fs(Example)i(primary)f(arra)o(y)f (image)h(header)-116 74 y Fp(SIMPLE)42 b(=)436 b(T)21 b(/)h(file)f(does)g(conform)f(to)h(FITS)g(standard)-116 124 y(BITPIX)42 b(=)414 b(32)21 b(/)h(number)e(of)i(bits)f(per)g(data)g (pixel)-116 174 y(NAXIS)64 b(=)436 b(2)21 b(/)h(number)e(of)i(data)f (axes)-116 224 y(NAXIS1)42 b(=)392 b(300)21 b(/)h(length)e(of)i(data)f (axis)64 b(1)-116 274 y(NAXIS2)42 b(=)392 b(200)21 b(/)h(length)e(of)i (data)f(axis)64 b(2)-116 373 y(CTYPE1)42 b(=)21 b('RA---TAN')238 b(/)22 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Fl(1)1395 2060 y(A)1438 1985 y(0)1438 2060 y(@)1486 2018 y Fo(l)1499 2003 y Fj(0)1474 2068 y Fo(m)1510 2053 y Fj(0)1480 2118 y Fo(n)1505 2103 y Fj(0)1533 1985 y Fl(1)1533 2060 y(A)1583 2068 y Fo(;)330 b Fw(\(D1\))970 2199 y(where)999 2282 y Fo(l)1012 2264 y Fj(0)1045 2282 y Fw(=)21 b(cos)7 b Fo(\016)i Fw(cos)f Fo(\013)976 2344 y(m)1012 2327 y Fj(0)1045 2344 y Fw(=)21 b(cos)7 b Fo(\016)i Fw(sin)e Fo(\013)988 2406 y(n)1013 2389 y Fj(0)1045 2406 y Fw(=)21 b(sin)7 b Fo(\016)1011 2468 y(l)22 b Fw(=)f(cos)7 b Fo(\022)i Fw(cos)e Fo(\036)988 2531 y(m)21 b Fw(=)g(cos)7 b Fo(\022)i Fw(sin)e Fo(\036)999 2593 y(n)21 b Fw(=)g(sin)7 b Fo(\022)970 2655 y(r)989 2661 y Fu(11)1045 2655 y Fw(=)21 b Fn(\000)7 b Fw(sin)g Fo(\013)1222 2661 y Fi(P)1256 2655 y Fw(sin)g Fo(\036)1339 2661 y Fi(P)1376 2655 y Fn(\000)i Fw(cos)f Fo(\013)1507 2661 y Fi(P)1541 2655 y Fw(cos)f Fo(\036)1628 2661 y Fi(P)1663 2655 y Fw(sin)g Fo(\016)1739 2661 y Fi(P)p eop %%Page: 40 40 40 39 bop -116 -192 a Fs(40)430 b(Greisen)14 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y(P)o(earson,)h(F.,)e(1984,)i Fq(Map)f(Pr)n(oje)n(ction)f(Metho)n(ds)p Fs(,)f(Sigma)j(Scien)o (ti\014c,)1024 1871 y(Inc.,)c(Blac)o(ksburg,)j(Virginia.)970 1916 y(P)o(onz,)e(J.)f(D.,)h(Thompson,)h(R.)e(W.,)h(Mu)q(~)-20 b(noz,)14 b(1994,)f(A&AS,)f(105,)h(53.)970 1962 y(W)m(ells,)18 b(D.)f(C.,)f(Greisen,)i(E.)e(W.,)h(Harten,)g(R.)f(H.,)g(1981,)h(FITS:)g (A)1024 2008 y(Flexible)f(Image)d(T)m(ransp)q(ort)g(System,)g(A&AS,)g (44,)g(363.)970 2602 y(This)f(article)g(w)o(as)f(pro)q(cessed)h(b)o(y)g (the)f(author)h(using)g(Springer-V)m(erlag)970 2647 y(L)980 2639 y Ft(A)997 2647 y Fs(T)1019 2655 y(E)1040 2647 y(X)g(A&A)g(st)o (yle)i(\014le)g Fq(L-AA)e Fs(v)o(ersion)i(3.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF