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["5181" "Tue" "2" "February" "1993" "23:49:33" "GMT" "Don Wells" "dwells at fits.CV.NRAO.EDU " nil "118" "Incunabula!" "^From:" nil nil "2"])
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From: dwells at fits.CV.NRAO.EDU (Don Wells)
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To: fitsbits at fits.CV.NRAO.EDU
Subject: Incunabula!
Date: Tue, 2 Feb 1993 23:49:33 GMT
Today Eric Greisen and I made a wonderful discovery. I told Eric that I
had found a tape of early FITS files from April 1980, and I said (sadly)
that this was the oldest FITS tape known to me. He then looked in his rack
of 9-track tapes and found a reel of tape whose typed label says:
"FITS Test Tape written at KPNO. unlabelled, 9-track,
1600bpi. 7 files, 5 written by program WFITS and 2
copied from Greisen's original test tape. Don Wells, 19Sep79"
This is the oldest FITS tape so far discovered. Also, the sixth file on it
is a copy of the first file on the first FITS tape, written in April 1979.
I have copied four of the files from the September 1979 tape and one file
from the April 1980 tape to directory /FITS/TestFiles/Incunabula on
anonymous-FTP server fits.cv.nrao.edu [192.33.115.8]. I append the README
from that directory.
I will be delighted to accept examples of early FITS files from other
observatories for installation in this directory.
----------------- /FITS/TestFiles/Incunabula/README ------------------
Incunabula
-=-=-=-=-=-=-
Don Wells
2-February-93
Incunabula noun pl. [1] The beginnings or the
earliest monuments of an art, race, or development;
cradle; birthplace. [2] {incunabulum, sing.}
Specifically, specimens of printing and
block-engraving that appeared before A.D. 1500.
The FITS files in directory /FITS/TestFiles/Incunabula were produced
early in the history of FITS, before the Basic_FITS paper was received
for publication by A&A Supplement Series on March 31, 1980. (The file
first-uv.fits has a DATE keyword only 12 days after this limit, and is
included because it is the oldest known example of the "Random Groups"
format.) Additional sample files with early values of DATE,
preferably before March 31, 1980 but certainly before June 1981
(publication of Basic_FITS), will be gratefully accepted for inclusion
in the directory.
filename bytes
------------- -------
first-uv.fits 25920
first.fits 1056960
m87-32.fits 267840
mndrll-8.fits 792000
orion-16.fits 529920
-=-=-=- first.fits -=-=-=-
This was the first file on the first FITS interchange tape. Eric
Greisen (NRAO) sent the tape to Don Wells (KPNO [now NOAO]). The tape
was produced by a PL/I program on an IBM_360 in April 1979 and was
read by a Fortran program on a CDC_6400 about a month later. That
first tape has since been lost, but the two files on it were copied by
Don Wells to another tape on September 19, 1979 and sent back to Eric
Greisen, who still possesses the copy tape. The tape has no parity
errors after 13.4 years. The data for this 5_GHz image of radio source
0810+665 were obtained while the VLA was still under construction.
Fewer than 27 antennas were available and the image was computed two
years before the invention of the self-calibration algorithm, so this
image is not representative of either the present VLA or the present
state of the art of synthesis imaging. The software used on the
IBM_360 was not AIPS, the design of which had only recently begun at
that time.
DATE-MAP= '16/10/78' / MAP CREATION DATE DD/MM/YY
DATE = '19/04/79' / MAP WRITING DATE DD/MM/YY
ORIGIN = 'NRAO(CV) PGM=DEC2FITS(V1)'
File first.lst is a listing of the header of first.fits as produced by
program "listfits".
-=-=-=- first-uv.fits -=-=-=-
This file is believed to be the oldest surviving example of the Random
Groups format used by radio synthesis imaging systems for about a
decade. Recently it has become common to use the BINTABLE extension
format for fringe visibility data instead of random groups.
DATE-OBS= '05/01/80' / OBSERVATION START DATE DD/MM/YY
DATE = '12/04/80' / TAPE WRITING DATE: DD/MM/YY
ORIGIN = 'NRAO(CV) PGM=DUV2FITS(V1)' /
-=-=-=- mndrll-8.fits -=-=-=-
This is believed to be the oldest file with BITPIX=8. It was also the
first image file on the first FITS tape sent from KPNO to NRAO.
ORIGIN = 'KPNO -- WFITS OF 09/13/79.' /
DATE = '13/09/79' '10.00.04' 'DCWGFAG' / DATE TIME JOBNAME
-=-=-=- orion-16.fits -=-=-=-
This is believed to be the first file with BITPIX=16 produced by KPNO.
ORIGIN = 'KPNO -- WFITS OF 09/13/79.' /
DATE = '13/09/79' '10.02.56' 'DCWGFAG' / DATE TIME JOBNAME
-=-=-=- m87-32.fits -=-=-=-
This is believed to be the oldest surviving file with BITPIX=32.
ORIGIN = 'KPNO -- WFITS OF 09/13/79.' /
DATE = '13/09/79' '10.03.51' 'DCWGFAG' / DATE TIME JOBNAME
The times on the DATE cards of the three successive files are
an indication of the speed of the batch job on the CDC_6400.
--
Donald C. Wells Associate Scientist dwells at nrao.edu
National Radio Astronomy Observatory +1-804-296-0277
520 Edgemont Road Fax= +1-804-296-0278
Charlottesville, Virginia 22903-2475 USA 78:31.1W, 38:02.2N
From fitsbits-request Wed Feb 3 10:09:07 1993
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["1482" "Wed" "3" "February" "93" "10:11:03" "EST" "Eric Greisen" "egreisen at primate.CV.NRAO.EDU " nil "31" "Correction of coordinate computation correction" "^From:" nil nil "2"])
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From: egreisen at primate.CV.NRAO.EDU (Eric Greisen)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at primate.CV.NRAO.EDU
Subject: Correction of coordinate computation correction
Date: Wed, 3 Feb 93 10:11:03 EST
A couple of days ago, I put out a message concerning the conversion
of the old coordinate nomenclature into the new (proposed) form.
In particular, I was concerned to point out that there is a formula
for this in the text of Chapter 4 of the NOST FITS Reference Guide
(which directly reflects a draft of the Hanisch/Wells paper on "World
Coordinate Systems") which I believe to be in error. Unfortunately,
the corrected formulae which I proposed turn out also to have an
error. I was confused by the complexities of the internal plot code
in AIPS. The actual formulae used by AIPS (and other packages) are
CDi_i = Di cos(rho)
CDj_j = Dj cos(rho)
CDi_j = -Dj sin(rho)
CDj_i = Di sin(rho)
where i refers to the longitude-like axis, j the latitude-like axis,
Dn = CDELTn, and rho = CROTAj. The old formulae for coordinates do
define the sign of rho implicitly, but I can't find it written down
anywhere. The formulae show that rho is the angle measured from the
+y axis to the +latitude axis (actually direction cosine axis in the
+latitude direction) in the clock sense toward the +x axis. Thus, the
angle is measured counter-clockwise for left-handed systems (our usual)
and clockwise for right-handed systems. Fortunately, the CD nomenclature
will eliminate rho and all this confusing sign business. (Of course,
we will undoubtedly find something else to confuse us instead.)
Eric W. Greisen, Scientist
National Radio Astronomy Observatory
Charlottesville, VA
From fitsbits-request Thu Feb 4 11:35:24 1993
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["1969" "Thu" "4" "February" "93" "11:35:50" "EST" "William Pence" "pence at tetra.gsfc.nasa.gov " nil "41" "New release of FITSIO V3.31" "^From:" nil nil "2"])
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To: fitsbits at fits.CV.NRAO.EDU, wgas at hypatia.gsfc.nasa.gov
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Subject: New release of FITSIO V3.31
Date: Thu, 4 Feb 93 11:35:50 EST
FITSIO - Version 3.31
4 February 1992
A new mini-release of the FITSIO subroutine library, version 3.31, is
now available. This new version contains several bug repairs and
efficiency improvements as well as an updated documentation file.
Otherwise, it is functionally identical and completely compatible with
the previous FITSIO release. See the 'release.doc' file in the FITSIO
distribution directory for further details on the changes made to this
version.
For those unfamiliar with FITSIO, it is a powerful yet simple to use
Fortran subroutine interface for reading and writing files in FITS
format on magnetic disk. It runs on most common types of computers and
supports FITS ASCII tables, binary tables, and image extensions as well
as simple array FITS files.
The FITSIO software, documentation, and example programs can be
obtained via anonymous ftp from:
tetra.gsfc.nasa.gov (128.183.8.77)
Type the following commands at the ftp prompt to copy any desired files:
ftp> user anonymous
Password: [type your username as a password]
ftp> cd pub/fitsio [to move to the fitsio subdirectory]
ftp> ls [to see a list of the available files]
ftp> get Read.me [contains latest information on FITSIO]
ftp> get fitsio.doc [complete user documentation]
ftp> get fitsio.tex [Latex version of the documentation]
ftp> get ... [get any additional desired files]
ftp> quit
-------------------------------------------------------------------------
Dr. William Pence
USRA/HEASARC (High Energy Astrophysics Science Archive Research Center)
Code 668
NASA/Goddard Space Flight Center SPAN: LHEAVX::PENCE
Greenbelt, MD 20771 Internet: pence at tetra.gsfc.nasa.gov
Telephone: (301)286-4599
From fitsbits-request Thu Feb 4 17:34:35 1993
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["4622" "Thu" "4" "February" "93" "17:36:30" "EST" "Eric Greisen" "egreisen at primate.CV.NRAO.EDU " nil "101" "WCS draft paper for comments" "^From:" nil nil "2"])
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From: egreisen at primate.CV.NRAO.EDU (Eric Greisen)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at primate.CV.NRAO.EDU
Subject: WCS draft paper for comments
Date: Thu, 4 Feb 93 17:36:30 EST
I have been working on coordinate representation in FITS and have,
with Mark Calabretta's great help, prepared a draft proposal. It
is just that - a draft. I'm sure I will find things I forgot to
put in as will Mark and I am sure that many of you will have
suggestions as well. I hope that you will keep them temperate and
constructive. Since I am about burned out on this for a few days
or more, I have decided to release the drafts for comment. They
are on the computer called fits in the anonymous ftp area with the
following README file. I look forward to hearing from you - I think.
Eric
C-----------------------------------------------------------------------
Representation of Coordinates in FITS
This anonymous ftp area (fits.cv.nrao.edu [192.33.115.8] :
/FITS/Documents/FITS_wcs) has been set aside for documents concerning
the representation of the coordinates of the pixels in FITS images.
My two early AIPS Memos (27 and 46) have been recovered and recast
into modern LaTeX (from TeX -1 or so). In so doing, we simply
digitized the old figures with a scanner and inserted them in the
documents. They are called aips27.ps.Z and aips46.ps.Z and are of
a reasonable size when compressed. They become over 2 and 1.4 Megabytes,
resp., when uncompressed! Try "zcat | lpr -Pxxx" to avoid on-disk
decompression; else "lpr -s -Pxxx " to avoid complaints about
too-large files.
The working document for a FITS paper authored by myself and Mark
Calabretta (and presumably others to be named later for political
or real reasons) is called wcs.ps.xxxx.Z. The xxxx is "none",
"some", "most", and "all" depending on the figures actually
represented in the paper (rather than shown as blank \vskip's).
The figures are drawn with postscript programs which have all been
verified with our printers and ghostview. However, we have found
several models of "postscript" printer which simply fail on the
documents and others which work, but are exceedingly slow. Hence,
the quicker-to-print versions.
If you find that you cannot read the papers via ftp, or cannot get
them to print, you may obtain paper copies of all 3 are by contacting
Ernest Allen
National Radio Astronomy Observatory
520 Edgemont Road
Charlottesville, VA 22903-2475
e-mail: eallen at nrao.edu
phone: 804-296-0209
fax: 804-296-0328
Please let us know your comments (well-considered and temperate only), so
that we may develop a consensus which will allow the final wcs.ps to
become a new FITS standard.
Thanks,
Eric W. Greisen,
Scientist.
National Radio Astronomy Observatory
egreisen at nrao.edu
804-296-0348
From fitsbits-request Wed Feb 10 11:58:07 1993
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["1174" "Wed" "10" "February" "93" "11:58:19" "EST" "William Pence" "pence at tetra.gsfc.nasa.gov " nil "25" "CFITSIO Beta Testers Wanted" "^From:" nil nil "2"])
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Subject: CFITSIO Beta Testers Wanted
Date: Wed, 10 Feb 93 11:58:19 EST
CFITSIO Beta Testers Wanted
We now have available a test version of CFITSIO, which is a C version
of the FITSIO subroutine interface for reading and writing files in
FITS format. We are looking for volunteer beta testers who would be
willing to try using this version for the next few weeks, prior to
the general release. If you are interested in this, please send
me a mail message, including which type of machine (e.g., SUN, VAX/VMS, etc.)
you will be using.
CFITSIO is not a complete re-write of FITSIO in C, but rather it
consists of a set of C wrapper routines which sit on top of the Fortran
FITSIO code. Thus, one must also have the lastest version of the
Fortran FITSIO library (Version 3.31 was released last week) to be able
to use CFITSIO. CFITSIO was written by Bruce O'Neel here at the
HEASARC and it uses the very useful CFORTRAN library written by
Burkhard Burrow at the U. of Toronto which provides a completely
transparent, machine independent interface between C and FORTRAN
routines. A copy of the CFORTRAN library is included with the CFITSIO
package.
William Pence
NASA/GSFC HEASARC
pence at tetra.gsfc.nasa.gov
LHEAVX::PENCE
From fitsbits-request Sat Feb 13 17:25:38 1993
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["434" "" "12" "February" "93" "07:44:08" "GMT" "Bruce Sams" "bruce at head-cfa.harvard.edu " nil "13" "are negative bitpix values legal??" "^From:" nil nil "2"])
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Organization: Smithsonian Astrophysical Observatory, Cambridge, MA, USA
Path: cv3.cv.nrao.edu!uvaarpa!darwin.sura.net!newsserver.jvnc.net!yale.edu!qt.cs.utexas.edu!news.Brown.EDU!noc.near.net!ceylon!hsdndev!cfa203!bruce
From: bruce at head-cfa.harvard.edu (Bruce Sams)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at fits.CV.NRAO.EDU
Subject: are negative bitpix values legal??
Date: 12 Feb 93 07:44:08 GMT
Hello there,
I'm writing a very simple FITS i/o facility for our data system here
at the Max Planck Institute in Munich, and I've come across a
question about whether it is legal to have negative values for BITPIX
in order to indicate that the dta is floating point rather than integer.
Can someone please tell me if this is legal? I have not seen it
before, but the SAOIMAGE fits reader seems to use it.
Many thanks,
Bruce Sams
From fitsbits-request Sat Feb 13 18:24:21 1993
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["1331" "Fri" "12" "February" "1993" "20:31:16" "GMT" "Orin Day" "oday at lobster.gsfc.nasa.gov " nil "32" "Re: are negative bitpix values legal??" "^From:" nil nil "2"])
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References: <1993Feb12.074408.17376 at cfa160.harvard.edu>
Reply-To: oday at lobster.gsfc.nasa.gov (Orin Day)
From: oday at lobster.gsfc.nasa.gov (Orin Day)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at fits.CV.NRAO.EDU
Subject: Re: are negative bitpix values legal??
Date: Fri, 12 Feb 1993 20:31:16 GMT
In article <1993Feb12.074408.17376 at cfa160.harvard.edu>
bruce at head-cfa.harvard.edu (Bruce Sams) writes:
> I'm writing a very simple FITS i/o facility for our data system here
> at the Max Planck Institute in Munich, and I've come across a
> question about whether it is legal to have negative values for BITPIX
> in order to indicate that the dta is floating point rather than integer.
> Can someone please tell me if this is legal?
Quoting directly the 3.0 NOST FITS User's Guide, p.16:
****************************************************************
2. BITPIX (integer) describes how an array value is represented:
8 --> ASCII or 8-bit unsigned integers
16 --> 16 bit, twos complement signed integers
32 --> 32 bit, twos complement signed integers
-32 --> IEEE 32-bit floating point values
-64 --> IEEE 64-bit floating point values
No other values for BITPIX are valid
****************************************************************
I believe the current standards document can be obtained via
anonymous FTP from nssdca.gsfc.nasa.gov.
OD
--
************* "How about a little fire, Scarecrow?" **************
Orin Day, Code 664.1 oday at lobster.gsfc.nasa.gov
Hughes STX/NASA GSFC Laboratory for High Energy Astrophysics
*******"Think that this is NASA's opinion? NO, NO, NO!!!" ******
From fitsbits-request Sat Feb 13 18:43:15 1993
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["811" "Fri" "12" "February" "1993" "21:17:00" "GMT" "Barry Schlesinger" "bschlesinger at nssdca.gsfc.nasa.gov " nil "22" "Re: are negative bitpix values legal??" "^From:" nil nil "2"])
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References: <1993Feb12.074408.17376 at cfa160.harvard.edu>
From: bschlesinger at nssdca.gsfc.nasa.gov (Barry Schlesinger)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at fits.CV.NRAO.EDU
Subject: Re: are negative bitpix values legal??
Date: Fri, 12 Feb 1993 21:17:00 GMT
In article <1993Feb12.074408.17376 at cfa160.harvard.edu>, bruce at head-cfa.harvard.edu (Bruce Sams) writes...
>I'm writing a very simple FITS i/o facility for our data system here
>at the Max Planck Institute in Munich, and I've come across a
>question about whether it is legal to have negative values for BITPIX
>in order to indicate that the dta is floating point rather than integer.
>Can someone please tell me if this is legal? I have not seen it
>before, but the SAOIMAGE fits reader seems to use it.
>
BITPIX = -32 means that the data are in single precision (32-bit) IEEE
floating point.
BITPIX = -64 means that the data following are in double precision
(64-bit) IEEE floating point.
No other negative values of BITPIX are permitted.
Barry Schlesinger
NSSDC/NOST
FITS Support Office
From fitsbits-request Wed Feb 17 10:59:24 1993
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["8598" "Wed" "17" "February" "1993" "15:26:00" "GMT" "Barry Schlesinger" "bschlesinger at nssdca.gsfc.nasa.gov " nil "167" "FITS basics and information (periodic posting)" "^From:" nil nil "2"])
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From: bschlesinger at nssdca.gsfc.nasa.gov (Barry Schlesinger)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at fits.CV.NRAO.EDU
Subject: FITS basics and information (periodic posting)
Date: Wed, 17 Feb 1993 15:26:00 GMT
for the benefit of new readers and the reference of old readers.
Message-ID: <17FEB199310264540 at nssdca.gsfc.nasa.gov>
Organization: NASA - Goddard Space Flight Center
News-Software: VAX/VMS VNEWS 1.41
FITS (Flexible Image Transport System) is a data format
designed to provide a means for convenient exchange of astronomical
data between installations whose standard internal formats and
hardware differ. A FITS file is composed of a sequence of Header Data
Units (HDUs). The header consists of keyword=value statements, which
describe the format and organization of the data in the HDU and may
also provide additional information, for example, about the instrument
or the history of the data. The data follow, structured as the header
specifies. The data section of the HDU may contain a digital image,
but, except for the first, IT DOESN'T HAVE TO. Other possible formats
include tables and multidimensional matrices that are not images. The
first HDU must contain a multidimensional matrix or no data at all;
the data in subsequent HDUs, called extensions, may be of any type,
consistent with certain rules. The "Image" in the name comes from the
original use of the format to transport digital images, but it's not
just for images any more.
FITS is not principally a graphics format designed for the
transfer of pictures; it does not incorporate "FITS viewers", packages
for decoding the data into an image. Users must develop or obtain
separate software to convert the data from the FITS file into a form
that can be readily displayed.
As has been discussed in this newsgroup, and in
alt.sci.astro.fits before it, the Extended Portable Bitmap Toolkit
(pbm+) can be used for converting many FITS files to such a format.
However, support is not guaranteed for all FITS files where the data
are in the form of an image. In particular, there may be problems
when the data matrix members are in IEEE floating point format
(BITPIX<0) or the matrix has more than two dimensions (NAXIS>2).
Archie Warnock and Ron Baalke have announced release of version
7.8 of the IMDISP program. IMDISP is an interactive image processing
program that runs on an IBM PC computer and supports FITS input.
IMDISP 7.8 is available via anonymous ftp at ames.arc.nasa.gov
[128.102.18.3] in a file called imdisp78.zip in the pub/SPACE/SOFTWARE
subdirectory and at hypatia.gsfc.nasa.gov in the pub/software/imdisp
subdirectory. It is also available through Simtel-20 [192.88.110.20]
at PD1:IMDISP78.ZIP.
Additional discussion of FITS->image converters appears in
this newsgroup from time to time.
The fundamental references on FITS are the following four
papers, often referred to collectively as the "Four FITS Papers".
These papers are the formal standard for FITS, endorsed by the
International Astronomical Union (IAU).
Wells, D. C., Greisen, E. W., and Harten, R. H., "FITS: a flexible
image transport system," Astronomy and Astrophysics Supplement Series,
44, 363-370, 1981.
Greisen, E. W. and Harten, R. H., "An extension of FITS for small
arrays of data," Astronomy and Astrophysics Supplement Series, 44,
371-374, 1981.
(NOTE: The format described in this paper has been used almost
exclusively to transport radio interferometry and is likely to be
replaced by other formats in the future. Writing data other than
radio interferometry data using this format is not recommended.)
Grosbol, P., Harten, R. H., Greisen, E. W., and Wells, D. C.,
"Generalized extensions and blocking factors for FITS," Astronomy and
Astrophysics Supplement Series, 73, 359-364, 1988.
Harten, R. H., Grosbol. P., Greisen, E. W., and Wells, D. C., "The
FITS tables extension, Astronomy and Astrophysics Supplement Series,
73, 365-372, 1988.
A User's Guide for FITS, commissioned by NASA Headquarters,
is maintained by the NASA/OSSA Office of Standards and Technology
(NOST) FITS Support Office. This Guide is intended to be a tutorial
for new FITS users. In addition to presenting the rules of FITS, it
provides some of the history and reasoning behind the choice of the
rules, adds recommendations on good practices, and discusses current
developments in FITS. A new version, 3.0, was issued in January 1993.
This document is at present available only in printed form, but steps
are under way to generate a PostScript version that will work on many
systems and a flat ASCII version.
NASA is sponsoring development of a formal standard for FITS.
The goal is a document codifying FITS as endorsed by the IAU,
eliminating some contradictions and ambiguities in the original FITS
papers, that can be endorsed by the IAU FITS Working Group as the FITS
standard. The document is being developed by a Technical Panel chaired
by Dr. Robert J. Hanisch (STSci), with review by the astronomical
community. Only minor revisions are expected to the current draft,
version 0.3b, but the form of the standard is not final, and it does
not supersede the four papers and Floating Point Agreement endorsed by
the IAU as the official standard for FITS.
The IAU has endorsed the Floating Point Agreement, which
defines how floating point numbers are to be expressed in FITS. The
basic agreement appears verbatim in the User's Guide, and the
substance is incorporated in the Draft NASA FITS definition.
The NOST maintains a file of FITS information available by
anonymous ftp from nssdca.gsfc.nasa.gov or DECnet copy from NSSDCA, in
the directory FITS. It includes copies of the current NASA draft
standard in flat ASCII, PostScript, and LaTeX. Style and index files
are provided for the LaTeX form. Except that the ASCII form does not
have an index, the standards in the three formats are identical; only
one need be retrieved. A current list of the extension type
(structure) names registered with the IAU FITS Working Group is
maintained. Also available, in LaTeX form, is the text of the
proposal for one of these new extension types, IMAGE. A README. file
describes the contents of the directory. A SOFTWARE subdirectory,
described by an included README.FIRST file, contains a program in C to
read and list the headers of a FITS file and another file with
information on publicly available FITS software packages. The ERRTEST
subdirectory contains several versions of the same FITS file, a valid
one and several with different kinds of header errors, for use in
testing software to read FITS files. An included README.FIRST file
contains details.
Additional material can be obtained by anonymous ftp from the
National Radio Astronomy Observatory, from fits.cv.nrao.edu, in the
directory FITS. The Documents subdirectory (case is significant)
contains copies of the BINTABLE Binary Tables extension proposal,
which is now under consideration by the FITS committees, and a draft
describing a proposed method for incorporating data compression under
FITS. It also contains text of the paper summarizing conclusions of a
workshop on World Coordinates held in Charlottesville in 1988 that is
serving as the basis for continuing discussion of world coordinates
issues, some of which appears on this newsgroup from time to time.
These documents are available in both LaTeX and PostScript forms. A
number of additional documents are available in ASCII text form,
including the proposal on physical blocking of FITS files on media
other than tape and material on FITS, its history, and the FITS
community.
Printed copies of many of the documents listed above can be
obtained from the NOST Librarian. Printed copies of the User's Guide
and either paper or electronic copies of the Draft NOST Standard, for
those without ftp access, are available. Because of restrictions set
by the copyright holder, NOST can send copies of the four FITS papers
only to non-profit organizations. The NOST can be reached as follows:
(Postal) NASA/OSSA Office of Standards and Technology
Code 633.2
Goddard Space Flight Center
Greenbelt MD 20771
USA
(Internet) nost at nssdca.gsfc.nasa.gov
(DECnet) NCF::NOST
Telephone: (301)286-3575 8 a. m. - 5 p. m., U. S. Eastern Time
If the Librarian is unavailable, a phone mail system takes the
call after four rings.
Please mention this posting in your request.
Use the FITS office electronic mail address below for replies
or questions. It is monitored by other NOST staff memebers when I am
away from the office and provides a greater certainty of a rapid
response.
Barry M. Schlesinger
Coordinator,
NASA/NSSDC
NOST FITS Support Office
(301) 513-1634 fits at nssdca.gsfc.nasa.gov
NCF::FITS
From dwells Thu Feb 18 12:59:39 1993
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From: fred at manono.span (Fred Patt (301)286-4569)
Sender: fitsbits-request at fits.CV.NRAO.EDU
Resent-From: dwells (Don Wells)
To: fitsbits
Subject: Comments on Draft WCS Standard
Date: Thu, 18 Feb 93 12:59:38 EST
Resent-Date: Thu, 18 Feb 93 12:59:38 EST
The following message was addressed to "fitsbits-request at nrao.edu",
rather than to "fitsbits at nrao.edu", and so it came to me, Don Wells,
the maintainer of the exploder, rather than being exploded as
intended.
------- Start of forwarded message -------
From: fred at manono.span (Fred Patt (301)286-4569)
To: fitsbits-request at fits.CV.NRAO.EDU
Subject: Comments on Draft WCS Standard
Date: Thu, 18 Feb 93 11:12:32 EST
Dr. Greisen,
I was offered a chance by Richard White to review your draft World
Coordinate System standard for FITS, due to my interest in map projections
in general and the quad sphere in particular. My recent involvement has
been with Earth science data sets but I was also involved with the COBE
mission for several years and am familiar with their skymap implementation.
I have advocated the adoption of the quad sphere by local Earth science
projects and was responsible for the use of this scheme in the Advanced
Very High Resolution Radiometer (AVHRR) Pathfinder Land prototype
processing system here at Goddard. I provided details of the projection
equations to the AVHRR group in the form of pseudocode, which has been
implemented in C by the developers. The data system manager, Mary James of
Goddard Code 902, has given me encouragement to publish my work on this
project.
My point here is that an exact form of the projection was derived
in the second of the two references (O'Neill and Laubscher, 1976) in terms
of polar coordinates, and this is the form which was implemented by AVHRR.
The reference only derived the projection in the forward direction, from
sky to cube coordinates, but I found that the derivation of the reverse
projection was very straightforward. I also found that the equations could
be significantly simplified by direct use of cartesian coordinates, thereby
eliminating the polar angles as intermediaries and, as a side benefit,
reducing the number of trig function evaluations required.
The information regarding the COBE implementation by Immanuel
Freedman is completely accurate. COBE has accepted the non-exact form of
the transformation, but I suggest that the description of the exact form is
at least as appropriate for your work. The combination of the overall
complexity of the projection equations and the errors in the projection
described for COBE would seem to discourage most potential users.
I have attached a set of summary notes which I provided for the
AVHRR group and the derivation of the reverse projection and cartesian form
equations for the exact transformation (in computer-language syntax; it
would be reasonable for me to put these into a more appropriate notation if
desired). I hope that this information may be of use to you.
Fred Patt
SUMMARY
The quadrilaterlized spherical cube, or quad sphere for short, is
an equal-area mapping and binning scheme for data collected on a spherical
surface (either Earth data or the celestial sphere). It was first proposed
by Chan and O'Neill in 1975 for the Naval Environmental Prediction Research
Facility (Reference 1).
There are two key elements to the quad sphere:
o The mapping consists of projecting the sphere onto the faces of an
inscribed cube using a curvilinear projection which preserves area. The
sphere is divided into six equal areas which correspond to the faces of the
cube. The vertices of the cube correspond to the cartesian coordinates
defined by |x|=|y|=|z| on the unit sphere. For an Earth projection, the
cube is normally oriented with one face normal to the North Pole and one
face centered on the Greenwich meridian (although any definition of pole
and meridian could be used). The faces of the cube are divided into square
bins, where the number of bins along each edge is a power of 2, selected to
produce the desired bin size. Thus the number of bins on each face is
2**(2*N), where N is the binning level, and the total number of bins is
6*2**(2*N). For example, a level of 10 gives 1024x1024 bins on each face
and 6291456 (6*2**20) total bins, which are 23.605 square arcminutes
(1.99737E-6 steradians) in size.
o The bins are numbered serially, rather than being rastered as for an
image. The bin numbers are determined as follows. The total number of
bits required for the bin numbers at level N is 2*N+3, where the 3 MSBs are
used for the face numbers and the remaining bits are used to number the
bins within each face. The faces are numbered 0-5 with 0 being the North
face, 1 through 4 being equatorial with 1 corresponding to Greenwich, and 5
being South. Thus at level 10, face 0 has bin numbers 0-1048577, face 1
has numbers 1048576-2097151, etc. Within each face the bins are numbered
serially from one corner (the convention is to start at the "lower left")
to the opposite corner, with the ordering such that each pair of bits
corresponds to a level of bin resolution. This ordering in effect is a
two-dimensional binary tree, which is referred to as the quad-tree The
conversion between bin numbers and coordinates is straightforward. If
4-byte integers are used for the bin numbers the maximum practical bin
level is 14, which uses 31 of the 32 bits and results in a bin size of
0.0922 square arcminutes (7.80223E-9 steradians). The advantages of this
numbering approach are stated below.
In principal the mapping and numbering schemes are separable; the
projection onto the cube could be used with another bin numbering scheme,
and the numbering scheme itself could be used with any arrangement of bins
which can be partitioned as a set of square arrays. Used together, they
comprise a flexible and efficient system for storing map data.
ADVANTAGES
The quad sphere projection does not produce singularities at the
poles or elsewhere, as do some other equal-area mapping schemes. The
distortion is moderate over the entire sphere, so that at no point are
shapes distorted beyond recognition.
Individual faces can be used to display parts of the sphere without
remapping by converting from pixel order to raster order; coordinate grids
can be overlaid on the faces to assist in registration.
The bin numbering allows the resolution to be changed (by factors
of two) simply by adding or deleting LSBs. This is particularly useful if
it is desired to increase the bin size, for example to compare maps with
different resolutions. This is performed simply by dividing the bin
numbers by four for each factor of two increase in bin size.
The bin numbers tend be closer together in sequence for neighboring
bins.
The quadtree bin numbering allows maps to be stored as FITS
BINTABLES with no wasted storage needed for blank areas (unfilled bins).
Only bins containing valid data need be included in a file, as the bin
numbers themselves identify the locations of the bins.
The serial numbering frees up one array dimension for another use
(e.g., time, a third coordinate, multichannel data, etc.).
DERIVATIONS OF REVERSE AND CARTESIAN TRANSFORMATIONS
The equations for the polar form of the transformation from the sphere to
the cube are given in reference 2 as equations (3-21) and (3-38). I have
taken this approach further in two areas: 1) inversion of these equations
to derive the cube-to-sphere transformation; and 2) simplification of these
equations to perform the transformation directly in terms of cartesian
coordinates. Both of these are straightforward as shown below.
1. DERIVATION OF THE CUBE-TO-SPHERE TRANSFORMATION
The exact form of the sphere-to-cube tranformation in reference 2 is
derived in terms of the co-elevation and azimuth angles (theta, phi) on the
sphere and an equivalent set of angles (mu,nu) on the cube. The equal-area
tranformation is performed using the following equations:
TAN(mu) = (12/pi)*(theta+ACOS(SIN(theta)*SIN(pi/4))-pi/2) (3-21)
TAN(nu) = SQRT((SEC(mu))**2*(1-COS(theta))/(1-COS(ATAN(SEC(theta))))) (3-38)
The inversion of these equations is as follows. Using the trig identity
ASIN(x)+ACOS(x)=pi/2, and the knowledge that SIN(pi/4) = 1/SQRT(2),
equation (3-21) can be written as
TAN(mu) = (12/pi)*(theta-ASIN(SIN(theta)/SQRT(2))) (3-21a)
and then rearranged as
theta - (pi/12)*TAN(mu) = ASIN(SIN(theta)/SQRT(2))
Taking the sine of both sides gives
SIN(theta)*COS((pi/12)*TAN(mu)) - COS(theta)*SIN((pi/12)*TAN(mu))
= SIN(theta)/SQRT(2)
Dividing through by COS(theta) and solving for TAN(theta) yields the
desired inverse relationship:
TAN(theta) = SIN((pi/12)*TAN(mu))/(COS((pi/12)*TAN(mu))-1/SQRT(2)) (1)
The inversion of equation (3-38) is performed by squaring both sides and
solving for COS(phi):
COS(phi) = 1 - (TAN(nu))**2*(1-COS(ATAN(SEC(theta))))/(SEC(mu)**2 (2)
As in the original derivation, this equation used the prior determination
of theta as a function of mu.
2. EXPRESSION OF THE TRANSFORMATIONS IN CARTESIAN COORDINATES
Both sets of transformation equations can be expressed in cartesian
coordinates (three-dimensional coordinates of the unit vector on the sphere
and two-dimensional coordinates on the cube face). Assume the unit vector
has coordinates (q,r,s) and the coordinates on the cube face are (x,y),
where q is normal to the plane of the cube face and (r,s) cooresponds to
(x,y). Then the functions of angles can be expressed as follows:
TAN(theta) = s/r
SIN(theta) = s/SQRT(r*r+s*s)
COS(phi) = q
TAN(mu) = y/x
TAN(nu) = SQRT(x*x+y*y)
Substituting these into the transformation equations results in the
following revised forms for the equations.
Equation (3-21a):
y/x = (12/pi)*(ATAN(s/r)- ASIN(s/SQRT(2*r*r+2*s*s))) (3)
Equation (3-38) can first be simplified by the following substitution:
COS(ATAN(SEC(theta))) = 1/SQRT(1+(SEC(theta))**2)
Substituting this and the appropriate cartesian coordinate expressions
gives:
x*x+y*y = (1+(y/x)**2)*(1-q)/(1-1/SQRT(2+(s/r)**2))
which can be re-written to eliminate y:
x = sqrt((1-q)/(1-1/SQRT(2+(s/r)**2))) (4)
Note that there is no ambiquity in the sign of the square root since the
derivation of (3-38) assumes that x is positive. This and equation (3) for
y/x above give the solution for both y and x, noting also that (3-21)
assumes that x is not less than y.
Equation (1) above can be rewritten also by direct substitution:
s/r = SIN((pi/12)*(y/x))/(COS((pi/12)*(y/x))-1/SQRT(2)) (5)
Equation (2) above can be rewritten by solving (4) for q:
q = 1 - x*x*(1-1/SQRT(2+(s/r)**2))) (6)
Thus equations (3) through (6) can be used to convert directly from
coordinates on the sphere to the cube and back without the need to convert
to angles as intermediate variables, and the number of trigonometric
function evaluations required is greatly reduced.
REFERENCES
1. Chan, F.K. and O'Neill (1975), Feasibility Study of a Quadrilateralized
Spherical Cube Earth Data Base, Computer Sciences Corporation, EPRF
Technical Report 2-75 (CSC). Prepared for the Environmental Prediction
Research Facility, Monterey, California.
2. O'Neill, I.M. and Laubscher, R.E. (1976), Extended Studies of a
Quadrilateralized Spherical Cube Earth Data Base, Computer Sciences
Corporation, NEPRF Technical Report 3-76 (CSC). Prepared for the Naval
Environmental Prediction Research Facility, Monterey, California.
------- End of forwarded message -------
From fitsbits-request Fri Feb 19 22:47:09 1993
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["1820" "" "19" "February" "1993" "22:54:57" "GMT" "Steve Allen" "sla at umbra.UCSC.EDU " nil "43" "little FITS tools" "^From:" nil nil "2"])
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From: sla at umbra.UCSC.EDU (Steve Allen)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at fits.CV.NRAO.EDU
Subject: little FITS tools
Date: 19 Feb 1993 22:54:57 GMT
In playing around with FITS files, I often want to do quick little
manipulations of them. I have several little shell scripts for
poking at them. I'm appending one at the end of this message.
(Apologies to VMS users, it's very Un*x. Also, nobody is allowed to make
rude comments about the little piece of Lick dirty laundry revealed here.)
Does anybody else have such little tools? Could we start a collection?
I call this "catfits"
---------------------------cut here-------------------------------------
#!/bin/csh
# A little Cshell script which acts like cat.
# Author: Steve Allen (sla at lick.ucsc.edu) 1992,1993
# It takes as input a FITS file, and cats out the headers of
# the primary HDU and any FITS extensions which may be present.
#
# Some recent Lick FITS headers have a "comment" as a part of the
# END card. This is actually illegal, but by giving the -L switch
# as the first argument this script will tolerate them.
if ( $1 == -L ) then
# be compatible with the illegal Lick FITS headers
set sed1 = '/^SIMPLE = /,/^END */p'
# throw away the -L
shift
else
# a normal, legal FITS header
set sed1 = '/^SIMPLE = /,/^END *$/p'
endif
set sed2 = '/^XTENSION= /,/^END *$/p'
#
if ( $#argv == 0 ) then
# act like a pipeline element
dd conv=unblock cbs=80 | sed -n -e "$sed1" -e "$sed2"
else
foreach file ($*)
if ($#argv > 1) echo "===> "$file" <==="
dd conv=unblock cbs=80 if=$file | sed -n -e "$sed1" -e "$sed2"
end
endif
_______________________________________________________________________________
Steve Allen | That was the equation! | sla at lick.ucsc.edu
UCO/Lick Observatory | Existence!...Survival must | If the UC were opining,
Santa Cruz, CA 95064 | cancel out programming! -- Ruk | it wouldn't use me.
From fitsbits-request Sat Feb 20 11:38:10 1993
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["1697" "Sat" "20" "February" "1993" "16:37:54" "GMT" "Don Wells" "dwells at fits.CV.NRAO.EDU " nil "38" "Re: little FITS tools" "^From:" nil nil "2"])
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From: dwells at fits.CV.NRAO.EDU (Don Wells)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at fits.CV.NRAO.EDU
Subject: Re: little FITS tools
Date: Sat, 20 Feb 1993 16:37:54 GMT
In article <1m3og1INN3vn at darkstar.UCSC.EDU> sla at umbra.UCSC.EDU (Steve
Allen) writes:
SA> In playing around with FITS files, I often want to do quick
SA> little manipulations of them. I have several little shell
SA> scripts for poking at them. I'm appending one at the end of this
SA> message.
This script is an elegant testimonial to the power of the Unix
shell(s) and tools. I have placed it in the anonymousFTP directory
fits.cv.nrao.edu [192.33.115.8]:/FITS/OSsupport/Unix/ as the file
-r--r--r-- 1 dwells 2131 Feb 20 11:00 catfits.csh
SA> .. nobody is allowed to make rude comments about the little piece
SA> of Lick dirty laundry revealed here..
In March 1979 Eric Greisen and I decided that END cards should have
trailing blanks in order to increase the probability that FITS-readers
would reliably detect the ends of headers; I myself have no regrets
about this design decision. Lick FITS-writer code is certainly not the
first to violate this rule. Practical implementations of FITS-readers
should be forgiving, of course.
SA> Does anybody else have such little tools? Could we start a
SA> collection?
I will be delighted to add such tools to the anonymousFTP server on
fits.cv.nrao.edu. I myself created one which does almost exactly the
same job as your "catfits"; it is in the same directory on the server:
-r--r--r-- 2 dwells 6300 Jul 28 1991 listfits.c
--
Donald C. Wells Associate Scientist dwells at nrao.edu
National Radio Astronomy Observatory +1-804-296-0277
520 Edgemont Road Fax= +1-804-296-0278
Charlottesville, Virginia 22903-2475 USA 78:31.1W, 38:02.2N
From fitsbits-request Sat Feb 20 16:57:07 1993
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["713" "Sat" "20" "February" "1993" "21:56:53" "GMT" "Don Wells" "dwells at fits.CV.NRAO.EDU " nil "14" "Re: little FITS tools" "^From:" nil nil "2"])
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From: dwells at fits.CV.NRAO.EDU (Don Wells)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at fits.CV.NRAO.EDU
Subject: Re: little FITS tools
Date: Sat, 20 Feb 1993 21:56:53 GMT
In article
dwells at fits.cv.nrao.edu (Don Wells) writes:
"... I myself created [a little FITS tool] which does almost exactly the
same job as your "catfits"; it is in the same directory on the server.."
I apologize to Barry Schlesinger. I neglected to mention that my
listfits.c is based directly on his program headlist.c available via
anonymous-FTP on host nssdca.gsfc.nasa.gov.
--
Donald C. Wells Associate Scientist dwells at nrao.edu
National Radio Astronomy Observatory +1-804-296-0277
520 Edgemont Road Fax= +1-804-296-0278
Charlottesville, Virginia 22903-2475 USA 78:31.1W, 38:02.2N
From fitsbits-request Mon Feb 22 09:13:41 1993
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["486" "Mon" "22" "February" "93" "09:15:11" "+0000" "nelson at axl.stsci.edu" "nelson at axl.stsci.edu" nil "16" "FITS tools \"catfits\" " "^From:" nil nil "2"])
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In-Reply-To: Your message of "19 Feb 93 22:54:57 GMT."
<1m3og1INN3vn at darkstar.UCSC.EDU>
X-Mts: smtp
From: nelson at axl.stsci.edu
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at fits.CV.NRAO.EDU
Cc: nelson at axl.stsci.edu
Subject: FITS tools "catfits"
Date: Mon, 22 Feb 93 09:15:11 +0000
Another 'catfits'; and I believe the first one with this name is the
catfits task in the IRAF/STSDAS fitsio package. Its features are as
follows:
1) Lists primary header and extension headers as well.
2) The user can choose from one line of information per header or a
full listing.
3) For the one liner, the user can tailor the keywords being displayed up to
132 columns per line. A log file can be created as well.
Nelson Zarate
Space Telescope Science Ins.
STSDAS group
From fitsbits-request Tue Feb 23 12:19:34 1993
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["295" "" "23" "February" "93" "10:40:20" "-0600" "hockey at iscsvax.uni.edu" "hockey at iscsvax.uni.edu" nil "4" "Coordinates from a Planetary Image?" "^From:" nil nil "2"])
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From: hockey at iscsvax.uni.edu
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at fits.CV.NRAO.EDU
Subject: Coordinates from a Planetary Image?
Date: 23 Feb 93 10:40:20 -0600
Does anybody know of an astronomical image-processing system that will permit
one to determine the latitude and longitude of a feature on a planetary image
(e. g., from the PDS Voyager spacecraft CD-ROM)? Neither IRAF nor MIRA have
this capability, and VICAR is dependent on obsolete hardware.
From fitsbits-request Tue Feb 23 22:18:25 1993
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["1260" "Tue" "23" "February" "1993" "19:13:24" "+0000" "Chris Marriott" "chris at chrism.demon.co.uk " nil "25" "Questions from a FITS novice" "^From:" nil nil "2"])
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Reply-To: chris at chrism.demon.co.uk
From: chris at chrism.demon.co.uk (Chris Marriott)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at fits.CV.NRAO.EDU
Subject: Questions from a FITS novice
Date: Tue, 23 Feb 1993 19:13:24 +0000
Please excuse me if the questions which follow are inappropriate for this
newsgroup. I know *nothing* about FITS, but I'd like to.
1. NASA produce a number of CD-ROMS containing images from space missions.
These files have a ".IMG" extension and appear to be a text header,
followed by binary data. Are these FITS files?
2. If the answer to the above question is "yes", can anyone give me advice
on where I could obtain information on the file format? The specific
reason I ask is that I am the author of a quite successful "shareware"
astronomy program for the IBM-PC, running under Microsoft Windows, and
I'd like to add the ability to display these .IMG files in the same
way the program can currently display GIF files, etc.
Thanks in advance for any help.
Chris
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From fitsbits-request Wed Feb 24 15:35:39 1993
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["817" "Wed" "24" "February" "93" "15:37:47" "EST" "Eric Greisen" "egreisen at primate.CV.NRAO.EDU " nil "18" "wcs paper" "^From:" nil nil "2"])
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From: egreisen at primate.CV.NRAO.EDU (Eric Greisen)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at primate.CV.NRAO.EDU
Subject: wcs paper
Date: Wed, 24 Feb 93 15:37:47 EST
I have received some useful comments and have folded them into the
paper and made a variety of other modifications. Several appendices
have been added (or are in progress), the quad-sphere description was
altered (to a Cobe quad sphere and an exact one), and numerous other
changes I can no longer remember. One of the figures (formerly on
page 28) had the chance to get an atan(0,0) in its postscript - so I
have corrected it to avoid the problem.
I encourage anyone with an interest in the problem to look at this
draft and forward comments to me or Mark Calabretta
(mcalabre at atnf.csiro.au). The papers are in the anonymous ftp area on
fits.cv.nrao.edu [192.33.115.8] known as /FITS/Documents/FITS_wcs and
there is a README file.
Thanks,
Eric W. Greisen, Scientist
National Radio Astronomy Observatory
From fitsbits-request Thu Feb 25 06:07:50 1993
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["10711" "Thu" "25" "February" "1993" "06:28:57" "GMT" "Mark Calabretta,123,412,6608870" "mcalabre at grus.csiro.au " nil "259" "WCS" "^From:" nil nil "2"])
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Reply-To: mcalabre at atnf.csiro.au
From: mcalabre at grus.csiro.au (Mark Calabretta,123,412,6608870)
Sender: fitsbits-request at fits.CV.NRAO.EDU
To: fitsbits at fits.CV.NRAO.EDU
Subject: WCS
Date: Thu, 25 Feb 1993 06:28:57 GMT
The appended latex document clarifies the relationship between the
elements of the CD matrix and the CDELTn and CROTAn parameters. It was
developed from a document by Bob Hanisch and Phil Hodge (STScI) in
consultation with Eric Greisen (NRAO) and will appear as an appendix of
the WCS document currently available for review from
fits.cv.nrao.edu:/FITS/Documents/FITS_wcs.
Mark Calabretta,
Australia Telescope
\documentstyle[11pt]{article}
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\begin{document}
\pagestyle{plain}
\thispagestyle{empty}
\begin{center} \large \bf
The CD Matrix in FITS and its Relationship to CDELTn and CROTAn \\
\normalsize \bf
\medskip
Mark Calabretta (ATNF) \\
Developed from a document by Bob Hanisch and Phil Hodge (ST ScI) \\
in consultation with Eric Greisen (NRAO) \\
\end{center}
The FITS {\tt CD} matrix describes a general linear transformation
from pixel coordinates $(i,j)$ to physical coordinates $(x,y)$, where
$(i,j)$ and $(x,y)$ are offsets from a reference point. In this
appendix we discuss the relationship of the {\tt CD} matrix to the
{\tt CDELT}{\it n} and {\tt CROTA}{\it n} parameters. In particular,
we will derive expressions for the elements of the {\tt CD} matrix in
terms of {\tt CDELT}{\it n} and {\tt CROTA}{\it n}.
Being a general linear transformation, the {\tt CD} matrix may
describe the geometrical operations of scaling, rotation and skewness.
Although scaling and rotation can be handled via {\tt CDELT}{\it n}
and {\tt CROTA}{\it n}, skewness cannot. Moreover, the {\tt
CROTA}{\it n} have generally been interpreted in a very limited way by
FITS readers. Specifically, only one {\tt CROTA}{\it n} parameter is
usually used, and it applies only to the two spherical coordinate axes
within the {\tt NAXIS}-dimensional FITS data cube.
A sequence of scaling and rotation operations sufficiently general for
astronomical purposes may be cast into matrix form as
\begin{equation}
\left[ \begin{array}{c} x \\
y \end{array} \right] =
\left[ \begin{array}{cc} S_x & 0 \\
0 & S_y \end{array} \right]
\left[ \begin{array}{cc} D_x & 0 \\
0 & D_y \end{array} \right]
\left[ \begin{array}{rc} \cos \theta & \sin \theta \\
-\sin \theta & \cos \theta \end{array} \right]
\left[ \begin{array}{cc} S_i & 0 \\
0 & S_j \end{array} \right]
\left[ \begin{array}{cc} D_i & 0 \\
0 & D_j \end{array} \right]
\left[ \begin{array}{c} i \\
j \end{array} \right] \,.
\end{equation}
This describes a scaling operation performed on the pixel coordinates,
followed by a rotation, followed by another scaling. Each of the
scaling matrices have been separated into two so that $D_i$, $D_j$,
$D_x$, and $D_y$ are positive quantities which describe the magnitude
of the scaling, and the $S_i$, $S_j$, $S_x$, and $S_y$ assume values
of $\pm 1$ and denote axis inversions. Note that, in general, the
operations of scaling and rotation do not commute.
In the context of optical astronomy, equation (1) might
correspond to the following physical situation: one has a telescope
with a scale at the focal plane given by $D_x$ and $D_y$, forming an
image on a detector; the detector is rotated with respect to $(x,y)$
by the angle $\theta$ and it has pixels of size $D_i$ by $D_j$ (in
units such as mm/pixel). $D_x$ and $D_y$ convert from mm on the
detector to the units of whatever is imaged onto the detector. For an
image of the sky, the units of $D_x$ and $D_y$ would be, for example,
degrees/mm, while for a long-slit spectrograph the units could be
\AA/mm for one axis and degrees/mm for the other. $S_i$ and $S_j$,
might assume values of $+1$ or $-1$ depending on whether the pixels
were read out of the detector in the forward or reverse direction, and
likewise the signs of $S_x$ and $S_y$ would depend on whether the
telescope optics had an even or odd number of reflections. In the
radioastronomical context, only the first scaling operation is
required so that $D_x = D_y = 1$.
The {\tt CD} matrix formed by combining the above scaling and rotation
matrices is
\begin{equation}
\left[ \begin{array}{c} x \\
y \end{array} \right] =
\left[ \begin{array}{rc} S_x D_x S_i D_i \cos \theta
& S_x D_x S_j D_j \sin \theta \\
-S_y D_y S_i D_i \sin \theta
& S_y D_y S_j D_j \cos \theta \end{array} \right]
\left[ \begin{array}{c} i \\
j \end{array} \right]
\end{equation}
whence
\begin{equation}
\begin{array}{lcr}
{\tt CD}i\_i & = & S_x D_x S_i D_i \cos \theta \,, \\
{\tt CD}i\_j & = & S_x D_x S_j D_j \sin \theta \,, \\
{\tt CD}j\_i & = & - S_y D_y S_i D_i \sin \theta \,, \\
{\tt CD}j\_j & = & S_y D_y S_j D_j \cos \theta \,.
\end{array}
\end{equation}
In restricted cases, it is possible to associate the elements of the
{\tt CD} matrix with the {\tt CDELT}{\it n} and {\tt CROTA}{\it n}
parameters. Consideration of the special case where the rotation
angle is zero allows us to identify the {\tt CDELT}{\it n} terms as
\begin{equation}
\begin{array}{lcr}
{\tt CDELT}i & = & S_x D_x S_i D_i \,, \\
{\tt CDELT}j & = & S_y D_y S_j D_j \,.
\end{array}
\end{equation}
However, if the rotation angle is non-zero, the matrix elements of
equation (2) generally cannot be expressed as functions
of the {\tt CDELT}{\it n} and {\tt CROTA}{\it n} and so constraints
must be sought. Possible constraints might be $D_i = D_j$ or $D_x =
D_y$. Of these, the latter is appropriate in the radioastronomical
context and it is also the more plausible in optical astronomy. (Note
that $D_x \neq D_y$ introduces a skew when the rotation angle is
not an integer multiple of $90^{\circ}$, a concept not supported in
the old notation.)
Now, since $S_x^2 = S_y^2 = +1$, we have
\begin{equation}
\begin{array}{lcl}
S_x S_j & = & S_x (S_y S_y) S_j \\
& = & (S_x S_y) \, (S_y S_j) \,.
\end{array}
\end{equation}
Likewise,
\begin{equation}
S_y S_i = (S_x S_y) \, (S_x S_i) \,. \\
\end{equation}
Then, with $D_x = D_y$, and noting that $S_x S_y = \pm 1$ so that
$\cos \theta = \cos (S_x S_y \theta)$ and $S_x S_y \sin \theta = \sin
(S_x S_y \theta)$, equations (3) and (4) may
be combined to yield
\begin{equation}
\begin{array}{lcr}
{\tt CD}i\_i & = & {\tt CDELT}i\, \cos(S_x S_y \theta) \,, \\
{\tt CD}i\_j & = & {\tt CDELT}j\, \sin(S_x S_y \theta) \,, \\
{\tt CD}j\_i & = & - {\tt CDELT}i\, \sin(S_x S_y \theta) \,, \\
{\tt CD}j\_j & = & {\tt CDELT}j\, \cos(S_x S_y \theta) \,.
\end{array}
\end{equation}
Likewise, equations (3) may be reduced to the form of
the Hanisch and Wells (1988) paper, by noting from
equations (4) that
\begin{equation}
\begin{array}{rcl}
\left | {\tt CDELT}i \right | & = & D_x D_i \\
& = & D_y D_i \,, \\
\left | {\tt CDELT}j \right | & = & D_y D_j \\
& = & D_x D_j \,, \\
{\rm sign}({\tt CDELT}i) & = & S_x S_i \,, \\
{\rm sign}({\tt CDELT}j) & = & S_y S_j \,,
\end{array}
\end{equation}
whence
\begin{equation}
\begin{array}{lcrcll}
{\tt CD}i\_i & = && {\tt CDELT}i
& \cos(S_i S_j \theta) \,, \\
{\tt CD}i\_j & = & {\rm sign} ( {\tt CDELT}i )
& \left | {\tt CDELT}j \right |
& \sin(S_i S_j \theta) \,, \\
{\tt CD}j\_i & = & - {\rm sign} ( {\tt CDELT}j )
& \left | {\tt CDELT}i \right |
& \sin(S_i S_j \theta) \,, \\
{\tt CD}j\_j & = && {\tt CDELT}j
& \cos(S_i S_j \theta) \,.
\end{array}
\end{equation}
Note that the arguments of the trigonometric functions in
equations (7) and (9) may differ in sign.
At this point we have yet to associate the {\tt CROTA}{\it n} with
$\theta$. The original FITS specification given by Wells, Greisen,
and Harten (1981) did not define how the {\tt
CROTA}{\it n} were to be interpreted but instead left each software
package to provide its own interpretation:
\begin{quote}
``The rotation parameters {\tt CROTA}{\it n} describe a coordinate
system which is rotated with respect to the normal (specified)
coordinate system. Users of this option should provide extensive
explanatory comments.''
\end{quote}
Consequently, different interpretations are possible for different
packages. It was stated above that, conventionally, only the {\tt
CROTA}{\it n} applying to the pair of spherical coordinate axes was
used. The scheme adopted by AIPS and other packages has
\begin{equation}
\begin{array}{lcl}
x & = & {\tt CDELT}i \,\cos ({\tt CROTA}j) \,i -
{\tt CDELT}j \,\sin ({\tt CROTA}j) \,j \,, \\
y & = & {\tt CDELT}i \,\sin ({\tt CROTA}j) \,i +
{\tt CDELT}j \,\cos ({\tt CROTA}j) \,j \,,
\end{array}
\end{equation}
where $i$ is the longitude-like axis and $j$ is the latitude-like
axis. Comparing these with equations (7) the
association
\begin{equation}
S_x S_y \theta = - {\tt CROTA}j
\end{equation}
may therefore be made, and the same conclusion can be drawn by
comparison with equations (9).
The final form of the equations for the elements of the {\tt CD}
matrix in terms of the {\tt CDELT}{\it n} and {\tt CROTA}{\it n} is
\begin{equation}
\begin{array}{lcr}
{\tt CD}i\_i & = & {\tt CDELT}i \,\cos({\tt CROTA}j) \,, \\
{\tt CD}i\_j & = & - {\tt CDELT}j \,\sin({\tt CROTA}j) \,, \\
{\tt CD}j\_i & = & {\tt CDELT}i \,\sin({\tt CROTA}j) \,, \\
{\tt CD}j\_j & = & {\tt CDELT}j \,\cos({\tt CROTA}j) \,.
\end{array}
\end{equation}
These equations may also be derived directly from
equations (9) by noting from equations (8)
and (11) that
\begin{equation}
\begin{array}{lcl}
S_i S_j \theta & = & S_i \,( S_x S_x )\, ( S_y S_y )\, S_j \,\theta \\
& = & ( S_i S_x )\, ( S_x S_y )\, ( S_y S_j )\, \theta \\
& = & - {\rm sign} ( {\tt CDELT}i ) \,
{\rm sign} ( {\tt CDELT}j ) \,{\tt CROTA}j \,.
\end{array}
\end{equation}
Although there was general agreement to support the conventions used
by AIPS in the NASA-sponsored conference of 1988 (Hanisch and Wells,
1988), the ambiguities in the original FITS papers mean
that there is no absolute guarantee that all packages have interpreted
the {\tt CROTA}{\it n} in accordance with equations (10).
\end{document}