WCS Paper II, revision
Mark Calabretta
Mark.Calabretta at atnf.csiro.au
Tue Apr 27 21:44:29 EDT 1999
Thank you to those who responded to the call for comments on WCS Paper II,
"Representations of celestial coordinates in FITS". A revised draft based
on this feedback and dated 1999/04/28 is now available from my home page
http://www.atnf.csiro.au/people/mcalabre
or from Eric's ftp area (soon, check the date)
ftp://ftp.cv.nrao.edu/NRAO-staff/egreisen.
The changes could probably best be described as fine tuning. "diff -C 1"
output on the LaTeX file is appended (insignificant typesetting changes have
been removed from this output). Changes were also made to correct keyword
nomenclature in Fig. 1 and to correctly identify the angle theta in Fig. 4.
There is still time to make further comments, particularly on Sect. 4.1.5,
Eqs. (40) & (41).
Mark Calabretta
ATNF
------------------------------------------------------------------------------
*** ccs.tex 1999/04/20 04:19:14 1.13
--- ccs.tex 1999/04/28 00:04:00
***************
*** 576,578 ****
the time of observation. The new keyword takes preference over the old one if
! both are given. Note that \EQUINOX{s} also applies to ecliptic as well as
equatorial coordinates.
--- 576,578 ----
the time of observation. The new keyword takes preference over the old one if
! both are given. Note that \EQUINOX{s} applies to ecliptic as well as to
equatorial coordinates.
***************
*** 834,839 ****
(ca.\,624-547\,{\sc b.c.}). The stereographic and orthographic date to
! Hipparchus (ca.\,190-after 126\,{\sc b.c.}).}, is widely used in optical
astronomy and was given its own code within the AIPS convention, namely
! \keyv{TAN}. However, the \keyv{TAN} projection is here extended to include
! corrections for instrumental distortions and is covered separately in
Sect.~\ref{sec:TAN}. If a simple gnomonic projection is required it can be
--- 834,841 ----
(ca.\,624-547\,{\sc b.c.}). The stereographic and orthographic date to
! Hipparchus (ca.\,190-after 126\,{\sc b.c.}).} is widely used in optical
astronomy and was given its own code within the AIPS convention, namely
! \keyv{TAN}\footnote{Referring to the dependence of $\Rt$ on the angular
! separation between the tangent point and field point, i.e. the native
! {\em co}-latitude.}. However, the \keyv{TAN} projection is here extended to
! include corrections for instrumental distortions and is covered separately in
Sect.~\ref{sec:TAN}. If a simple gnomonic projection is required it can be
***************
*** 899,903 ****
widely used in aperture synthesis radioastronomy and was given its own code
! within the AIPS convention, namely \keyv{SIN}. Use of this projection code
! obviates the need to specify an infinite value as a parameter of \keyv{AZP}.
! In this case, Eq.~(\ref{eq:AZPRt}) becomes
--- 901,906 ----
widely used in aperture synthesis radioastronomy and was given its own code
! within the AIPS convention, namely \keyv{SIN}\footnote{Similar etymology to
! \keyv{TAN}.}. Use of this projection code obviates the need to specify an
! infinite value as a parameter of \keyv{AZP}. In this case,
! Eq.~(\ref{eq:AZPRt}) becomes
***************
*** 1086,1108 ****
high-precision with a measuring engine. These plate coordinates are usually
! measured in $\mu$m rather than degrees.
! Astrometrists refer to the $(\xi,\eta)$ coordinates obtained by application of
! the distortion function as {\em standard} coordinates. They correspond to
! projection plane coordinates for a non-distorted projection, usually gnomonic
! or zenithal equidistant. We will confine ourselves here to the gnomonic case
! since the field sizes of interest are sufficiently small, typically less than
! $6\degr \times 6\degr$, that the distortion function itself can account for
! the difference between it and other projections.
!
! In the current treatment we require that $(\xi,\eta)$ be measured in degrees.
! However, in order to more easily accomodate traditional astrometric practices
! and facilitate translation of existing plate measurements, we recommend but do
! not require that $(x,y)$ be measured in degrees. This is possible because the
! linear terms of the distortion function provide an opportunity to rescale
! $(\xi,\eta)$ appropriately. The distortion function effectively becomes an
! attachment of the the linear transformation matrix rather than part of the
! projection itself. Ideally the linear terms of the distortion function should
! define an identity transformation with translation, rotation, skewness and
! scale handled solely by Eq.~(\ref{eq:ijCDxy}). In that case the distortion
! function becomes a second-order correction.
--- 1089,1106 ----
high-precision with a measuring engine. These plate coordinates are usually
! measured in $\mu$m or mm rather than degrees.
! The {\em standard} coordinates, $(\xi,\eta)$, correspond to projection plane
! coordinates for a gnomonic projection (Murray \cite{kn:Mu}). Standard
! coordinates are usually measured in radians but in the current treatment we
! require that $(\xi,\eta)$ be measured in degrees. However, in order to more
! easily accomodate traditional astrometric practices and facilitate translation
! of existing plate measurements, we recommend but do not require that $(x,y)$
! be measured in degrees. This is possible because the linear terms of the
! distortion function provide an opportunity to rescale $(\xi,\eta)$
! appropriately. The distortion function effectively becomes an attachment of
! the the linear transformation matrix rather than part of the projection
! itself. Ideally the linear terms of the distortion function should define an
! identity transformation with translation, rotation, skewness and scale handled
! solely by Eq.~(\ref{eq:ijCDxy}). In that case the distortion function becomes
! a second-order correction.
***************
*** 1396,1403 ****
! Lambert's\footnote{Johann Heinrich Lambert (1728-1777) formulated and gave his
! name to a number of important projections as listed in
! Table~\ref{ta:aliases}.} equal area projection, the case with $\lambda = 1$,
! is illustrated in Fig.~\ref{fig:CEA}. It shows the extreme compression of the
! parallels of latitude at the poles typical of all cylindrical equal area
! projections.
--- 1394,1401 ----
! Lambert's\footnote{The mathematician, astronomer and physicist Johann Heinrich
! Lambert (1728-1777) formulated and gave his name to a number of important
! projections as listed in Table~\ref{ta:aliases}.} equal area projection, the
! case with $\lambda = 1$, is illustrated in Fig.~\ref{fig:CEA}. It shows the
! extreme compression of the parallels of latitude at the poles typical of all
! cylindrical equal area projections.
***************
*** 1991,1993 ****
and use the fact that for a cone tangent to the sphere at latitude $\theta_1$
! as shown in Fig.~\ref{fig:COP} then $R_{\theta_1} = \rd\cot\theta_1$.
--- 1989,1991 ----
and use the fact that for a cone tangent to the sphere at latitude $\theta_1$
! as shown in Fig.~\ref{fig:COP} we have $R_{\theta_1} = \rd\cot\theta_1$.
***************
*** 2653,2658 ****
! However, FITS-writers should never mix \CDELT{i} or \CROTA{j} together with
! the new keyword values. We make this requirement primarily to minimize
! confusion. However, there is the practical case described in
! Sect.~\ref{sec:AITGLSMER} where the presence of \CDELT{j} header cards
distinguishes between the old and new interpretations of the \keyv{AIT},
--- 2651,2657 ----
! Modern FITS-writers must not attempt to help older interpreters by mixing
! \CDELT{i} or \CROTA{j} together with the new keyword values (assuming the
! \CD{j}{is} matrix is amenable to such translation). We make this requirement
! primarily to minimize confusion although there is the practical case described
! in Sect.~\ref{sec:AITGLSMER} where the presence of \CDELT{j} header cards
distinguishes between the old and new interpretations of the \keyv{AIT},
***************
*** 2833,2839 ****
optical telescopes is closer to a gnomonic (\keyv{TAN}) projection. On the
! other hand, the optics of the telescope used for the Sloan Digital Sky Survey
! produce a cylindrical perspective projection. Similarly a map of the surface
! of the moon as it appears from Earth requires a zenithal perspective
! (\keyv{AZP}) projection with $\mu = -60$. Likewise for spacecraft generated
! images of distant moons and planets.
--- 2832,2838 ----
optical telescopes is closer to a gnomonic (\keyv{TAN}) projection. On the
! other hand, the great circle scanning technique of the the Sloan Digital Sky
! Survey produce a cylindrical projection. Similarly a map of the surface of
! the moon as it appears from Earth requires a zenithal perspective (\keyv{AZP})
! projection with $\mu \approx -60$. Likewise for spacecraft generated images
! of distant moons and planets.
***************
*** 2915,2917 ****
\keyv{STG}, \keyv{ARC}, \keyv{ZEA}, \keyv{CAR}, \keyv{MER}, \keyv{BON},
! \keyv{PCO}, \keyv{GLS}, \keyv{AIT}. The following are scaled true at the
reference point for the particular conditions indicated: \keyv{TAN}
--- 2915,2917 ----
\keyv{STG}, \keyv{ARC}, \keyv{ZEA}, \keyv{CAR}, \keyv{MER}, \keyv{BON},
! \keyv{PCO}, \keyv{SFL}, \keyv{AIT}. The following are scaled true at the
reference point for the particular conditions indicated: \keyv{TAN}
***************
*** 3106,3109 ****
\verb+CD1_1A = -0.005 / Transformation matrix element+ \\
! \verb+CD1_2A = 0.00001 / Transformation matrix element+ \\
! \verb+CD2_1A = 0.00001 / Transformation matrix element+ \\
\verb+CD2_2A = 0.005 / Transformation matrix element+ \\
--- 3106,3109 ----
\verb+CD1_1A = -0.005 / Transformation matrix element+ \\
! \verb+CD1_2A = 0.00002 / Transformation matrix element+ \\
! \verb+CD2_1A = -0.00001 / Transformation matrix element+ \\
\verb+CD2_2A = 0.005 / Transformation matrix element+ \\
***************
*** 3358,3360 ****
$\phi\sub{p}$ to give $(\ell,b)$ in terms of $(\phi,\theta)$. This is easy
! because we know $\ell\sub{p}=90\degr$ and the simple special case
Eqs.~(\ref{eq:nat2std90}) apply so
--- 3354,3356 ----
$\phi\sub{p}$ to give $(\ell,b)$ in terms of $(\phi,\theta)$. This is easy
! because we know $b\sub{p}=90\degr$ and the simple special case
Eqs.~(\ref{eq:nat2std90}) apply so
***************
*** 3404,3406 ****
\noindent
! Equations~(\ref{eq:nat2stdm90}) apply for $\ell\sub{p}=-90\degr$ so
--- 3400,3402 ----
\noindent
! Equations~(\ref{eq:nat2stdm90}) apply for $b\sub{p}=-90\degr$ so
***************
*** 3597,3599 ****
approximation since typically $B_1 \approx A_1$. On the other hand, setting
! $\lambda = 1$ preserves $(x,y)$ as plate coordinates measured in mm.
--- 3593,3596 ----
approximation since typically $B_1 \approx A_1$. On the other hand, setting
! $\lambda = 1$ preserves $(x,y)$ as plate coordinates measured in mm. The
! choice will be left open here.
***************
*** 3690,3693 ****
& \keyw{PV1\_14} &=& a_{14} &=& 0 \, , \\
! & \keyw{PV1\_15} &=& a_{15} &=& 0 \, , \\
! & \keyw{PV1\_16} &=& a_{16} &=& 0 \, .
\end{array}
--- 3687,3689 ----
& \keyw{PV1\_14} &=& a_{14} &=& 0 \, , \\
! & \keyw{PV1\_15} &=& a_{15} &=& 0 \, .
\end{array}
***************
*** 3714,3717 ****
& \keyw{PV2\_14} &=& b_{14} &=& 0 \, , \\
! & \keyw{PV2\_15} &=& b_{15} &=& 0 \, , \\
! & \keyw{PV2\_16} &=& b_{16} &=& 0 \, .
\end{array}
--- 3710,3712 ----
& \keyw{PV2\_14} &=& b_{14} &=& 0 \, , \\
! & \keyw{PV2\_15} &=& b_{15} &=& 0 \, .
\end{array}
***************
*** 3792,3793 ****
--- 3787,3803 ----
! Most optical telescopes are better described by the \keyv{TAN} projection. In
! this case the only difference in the header would be
! \noindent
! \begin{eqnarray*}
! \keyw{CTYPE2} & = & \keyv{'RA---TAN'} \, , \\
! \keyw{CTYPE3} & = & \keyv{'DEC--TAN'} \, , \\
! \end{eqnarray*}
! \noindent
! Verifying it with the same values as before yields the same $(s,x,y)$ and thus
! the same wavelength. From Eqs.~(\ref{eq:azrt}), (\ref{eq:azphi}) and
! (\ref{eq:GNORt}) we get $(\phi,\theta) = (90\degr,89\fdg4314076)$, the offset
! of 2046\farcs933 differing by only 67~mas. The right ascension and
! declination are $(\alpha,\delta) = (150\fdg3449926,-34\fdg5070956)$.
***************
*** 4303,4305 ****
described in this paper. One, Mollweide's, even requires solution of a
! transcentental equation for the forward equations. However, these do not
give rise to any particular difficulties.
--- 4314,4316 ----
described in this paper. One, Mollweide's, even requires solution of a
! transcendental equation for the forward equations. However, these do not
give rise to any particular difficulties.
***************
*** 4355,4357 ****
Sinusoidal & \keyv{SFL} \\
! \In Global sinusoid \\
\In Mercator equal-area \\
--- 4366,4368 ----
Sinusoidal & \keyv{SFL} \\
! \In Global sinusoid (\keyv{GLS}) \\
\In Mercator equal-area \\
***************
*** 4500,4501 ****
--- 4511,4515 ----
+ \bibitem[1983]{kn:Mu} Murray, C. A., 1983, Vectorial Astrometry,
+ Adam Hilger Ltd., Bristol.
+
\bibitem[1976]{kn:OL} O'Neill, E. M., Laubscher, R. E., 1976,
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