Index: trunk/doc/pslib/psLibADD.tex =================================================================== --- trunk/doc/pslib/psLibADD.tex (revision 1560) +++ trunk/doc/pslib/psLibADD.tex (revision 1583) @@ -1,3 +1,3 @@ -%%% $Id: psLibADD.tex,v 1.28 2004-08-17 03:30:07 eugene Exp $ +%%% $Id: psLibADD.tex,v 1.29 2004-08-19 02:45:14 eugene Exp $ \documentclass[panstarrs]{panstarrs} @@ -897,4 +897,60 @@ \subsubsection{Celestial Coordinate Conversions} +Changes between spherical coordinate systems (ie, Ecliptic, Galactic, +and ICRS, or Celestial, coordinates) is equivalent to a rotation in +3D. Given two coordinate system, $\alpha,\delta$ and $\phi,\theta$ +which differ by only a rotation, the transformation between these two +systems are defined by the following three parameters: +\begin{itemize} +\item $\alpha_p$ : the longitude of the target system pole in the + source system +\item $\delta_p$ : the latitutde of the target system pole in the + source system. +\item $\phi_p$ : the longitude of the ascending node in the target system +\end{itemize} +Note that $\theta_p$, the latitude of the source system pole in the +target system, is equal to $\delta_p$ by symmetry. + +The relevant trigonometric relationships are: +\begin{eqnarray} +\sin \theta = \cos \delta \sin \delta_p \sin (\alpha - \alpha_p) + \sin \delta \cos \delta_p +\cos \theta \sin (\phi - \phi_p) = \cos \delta \cos \delta_p \sin (\alpha - \alpha_p) - \sin \delta \sin \delta_p +\cos \theta \cos (\phi - \phi_p) = \cos \delta \cos (\alpha - \alpha_p) +\end{eqnarray} +% +and for the inverse transformations, the equivalent relationships are: +% +\begin{eqnarray} +\sin \delta & = & \cos \theta \sin \delta_p \sin (\phi - \phi_p) + \sin \theta \cos \delta_p \\ +\cos \delta \sin (\alpha - \alpha_p) & = & \cos \theta \cos \delta_p \sin (\phi - \phi_p) - \sin \theta \sin \delta_p \\ +\cos \delta \cos (\alpha - \alpha_p) & = & \cos \theta \cos (\phi - \phi_p) +\end{eqnarray} +% +Since $\theta$ and $\delta$ have domains of $-\pi/2, \pi/2$, the value +of these angles are found by applying the arcsin to the sine of these +angles ($\theta = \arcsin \sin \theta$) which is always single-valued +and defined. The value of $\alpha$ (or $\phi$) is found from +\code{atan2(y,x)}, where $y = \cos \delta \sin (\alpha - \alpha_p)$ and +$x = \cos \delta \cos (\alpha - \alpha_p)$. + +\paragraph{Galactic to ICRS} + +\tbd{clean up these values - the transformations above need to be + checked for pole vs node coords} + +\begin{verbatim} +* P = 192.25 RA of galactic north pole (mean B1950.0) +* Q = 62.6 inclination of galactic to mean B1950.0 equator +* R = 33 longitude of ascending node + +We should precess L2,B2 coords from B1950 to epoch of interest + +265.600000 -28.916667 (B1950) +192.250000 27.400000 (B1950) + +266.394165 -28.936098 (J2000) +192.859536 27.128309 (J2000) +\end{verbatim} + These will be implemented using the corresponding SLALIB functions: @@ -912,47 +968,138 @@ \subsubsection{Projections} -The following information is from +We implement three types of projections: {\em zenithal}, {\em +cylindrical} and {\em pseudocylindrical}, each requiring slightly +different handling. Our representations are based on the treatment of +projections presented by \href{http://www.cv.nrao.edu/fits/documents/wcs/wcs.all.ps}{Greisen \& -Calabretta (1995, ADASS, 4, 233)}. - -Let the latitude be $\phi$ and the longitude $\theta$. The domains of -these are $-\pi < \phi \le \pi$ and $-\pi/2 \le \theta \le \pi/2$. - -For zenithal projections (e.g.\ Gnomonic and Orthographic) the -following hold: - -\begin{eqnarray} -x & = & R \sin (\phi) \\ -y & = & -R \cos (\phi) -\end{eqnarray} - -and - -\begin{eqnarray} -R & = & \sqrt{x^2 + y^2} \\ -\phi & = & {\rm arg} (-y,x) -\end{eqnarray} - -\paragraph{Gnomonic} - -The Gnomonic projection (``TAN'') is a zenithal projection. - -\begin{eqnarray} -R & = & \cot (\theta) 180^\circ/\pi \\ -\theta & = & \arctan (180^\circ/(\pi R)) -\end{eqnarray} - -\paragraph{Orthographic} - -The Orthographic projection (``SIN'') is a zenithal projection. - -\begin{eqnarray} -R & = & \cos (\theta) 180^\circ/\pi \\ -\theta & = & \arccos (\pi R / 180^\circ) -\end{eqnarray} - -\paragraph{Cartesian} - -The Cartesian projection (``CAR'') is a very simple cylindrical projection. +Calabretta (1995, ADASS, 4, 233)}. In all of these projections, we +are converting from a spherical coordinate $\alpha,\delta$ to a linear +(2-D) coordinate $x_p,y_p$. The projection is defined by the +projection type, the projection center ($\alpha_p, \delta_p$) and the +the plate scales in the $x_p$ and $y_p$ directions ($\rho_x,\rho_y$). + +In the structure, \code{psProjection}, the projection type is defined +by the element \code{type}, the projection center $\alpha_p,\delta_p$ +is defined by the elements \code{R,D}, and the plate scales, +$\rho_x,\rho_y$, are defined by the elements \code{Xs,Ys}. The plate +scales are applied independently to the $x$ and $y$ coordinates to +convert them to the corresponding linear units (ie, pixels): +% +\begin{eqnarray} +x_p & = & \rho_x x \\ +y_p & = & \rho_y y \\ +\end{eqnarray} +% +In the discussions below, we ignore this last step (or first step, +depending on the direction of the conversion). + +\paragraph{Zenithal Projections} + +The {\em zenithal} projections are defined relative to a set of +spherical coordinates with pole at the center of the projection +($\alpha_p, \delta_p$), and which thus represents a coordinate system +rotated relative to the coordinate system of $\alpha, \delta$. In +this spherical coordinate system, the coordinate of longitude is +labeled $\phi$, and has domain of $-\pi < \phi \le \pi$, while the +latitude, measured from the pole, is labeled $\theta$ and has domain +$0 \le \theta \le \pi$. The coordinate frame of $\phi,\theta$ is +defined so that $\phi_p$, the longitude of the target system pole, is +0.0. + +For an arbitrary projection center, it is necessary to convert the +spherical coordinates to be projected ($\alpha,\delta$) to the +projection spherical coordinate system coordinates ($\phi, \theta$). +In practice, we construct the following useful trigonometric +relationships between $\phi$ and $\theta$ which may be employed in the +equations of $x,y$ below: +% +\begin{eqnarray} +\sin \theta & = & \sin \delta \sin \delta_p + \cos \delta \cos \delta_p \cos (\alpha - \alpha_p) \\ +\cos \theta \cos \phi & = & \sin \delta \cos \delta_p - \cos \delta \sin \delta_p \cos (\alpha - \alpha_p) \\ +\cos \theta \sin \phi & = & - \cos \delta \sin (\alpha - \alpha_p) +\end{eqnarray} +% +For the inverse transformations, the equivalent relationships are: +% +\begin{eqnarray} +\sin \delta & = & \sin \theta \sin \delta_p + \cos \theta \cos \delta_p \cos \phi \\ +\cos \delta \cos (\alpha - \alpha_p) & = & \sin \theta \cos \delta_p - \cos \theta \sin \delta_p \cos \phi \\ +\cos \delta \sin (\alpha - \alpha_p) & = & - \cos \theta \sin \phi +\end{eqnarray} +% +For zenithal projections, the linear coordinates are related to +$\phi,\theta$ by: +% +\begin{eqnarray} +x & = & R_\theta \sin \phi \\ +y & = & -R_\theta \cos \phi +\end{eqnarray} +% +and the inverse: +% +\begin{eqnarray} +R_\theta & = & \sqrt{x^2 + y^2} \\ +\phi & = & {\rm arg} (-y,x) +\end{eqnarray} +% +The coordinates $x,y$ above are defined to be in angular units (ie, +radians). + +From these relationships, we can calculate $\alpha, \delta$ as: +% +\begin{eqnarray} +\alpha - \alpha_p & = & \arctan (\sin \alpha, \cos \alpha) \\ +\delta & = & \arcsin (\sin \delta) \\ +\end{eqnarray} +% +Note that if $(x,y) = (0,0)$, then $\alpha = \alpha_p, \delta = \delta_p$. + +\subparagraph{Gnomonic} + +The Gnomonic projection (``TAN'') is a zenithal projection with +$R_\theta = \cot \theta$. The resulting relationships for $(x,y)$ and +for $\sin \theta, \cos \theta$ are: + +\begin{eqnarray} +x & = & \frac{\cos \theta \sin \phi}{\sin \theta} \\ +y & = & \frac{-\cos \theta \cos \phi}{\sin \theta} \\ +\sin \theta & = & \zeta / \sqrt{1 + \zeta^2} \\ +\cos \theta & = & 1 / \sqrt{1 + \zeta^2} \\ +\end{eqnarray} + +where $\zeta = 1 / R_\theta$. + +\subparagraph{Orthographic} + +The Orthographic projection (``SIN'') is a zenithal projection with +$R_\theta = \cos \theta$. The resulting relationships for $(x,y)$ and +for $\sin \theta, \cos \theta$ are: + +\begin{eqnarray} +x & = & \cos \theta \sin \phi \\ +y & = & -\cos \theta \cos \phi \\ +\sin \theta & = & \sqrt{1 - R_\theta^2} \\ +\cos \theta & = & R_\theta \\ +\end{eqnarray} + +\paragraph{Cylindrical and Pseudocylindrical Projections} + +The {\em cylindrical} and {\em pseudocylindrical} projections are +defined relative to a set of cylindrical coordinates whose pole is +coincident with the pole of the spherical coordinates. These +projections are particularly used for full-sky representations, and +are only defined for projection centers with $\delta_p = 0$. In this +spherical coordinate system, the coordinate of longitude is labeled +$\phi$, and has domain of $-\pi < \phi \le \pi$, while the latitude, +measured from the pole, is labeled $\theta$ and has domain $0 \le +\theta \le \pi$. The projection center longitude, $\alpha_p$ +corresponds to $\phi = 0$, thus the value of $\phi$ is determined as +$\alpha - \alpha_p$ for all such projections. + +\subparagraph{Cartesian} + +The Cartesian projection (``CAR'') is a very simple cylindrical +projection with the following relationships between $x,y$ and +$\phi,\theta$: \begin{eqnarray} @@ -961,5 +1108,5 @@ \end{eqnarray} -\paragraph{Mercator} +\subparagraph{Mercator} The Mercator projection (``MER'') is a cylindrical projection. @@ -967,16 +1114,16 @@ \begin{eqnarray} x & = & \phi \\ -y & = & \ln \left( \tan (45^\circ + \theta/2) \right) 180^\circ/\pi \\ -{\rm and}\hspace{1cm} \theta & = & 2 \arctan \left( e^{y\pi/180^\circ} \right) - 90^\circ -\end{eqnarray} - -\paragraph{Hammer-Aitoff} - -The Hammer-Aitoff projection is a general projection, and is defined: - -\begin{eqnarray} -x & = & 2 \alpha \cos (\theta) \sin (\phi/2) \\ -y & = & \alpha \sin \theta \\ -{\rm where}\hspace{1cm} \alpha^{-1} & \equiv & (180^\circ/\pi) \sqrt{\left(1 + \cos (\theta) \cos (\phi/2) \right) / 2} +y & = & \ln \left( \tan (\pi/4 + \theta/2) \right) \\ +{\rm and}\hspace{1cm} \theta & = & 2 \arctan \left( e^y \right) - \pi/2 +\end{eqnarray} + +\subparagraph{Hammer-Aitoff} + +The Hammer-Aitoff projection(``AIT'') is a pseudocylindrical projection, and is defined: + +\begin{eqnarray} +x & = & 2 \zeta \cos \theta \sin \frac{\phi}{2} \\ +y & = & \zeta \sin \theta \\ +{\rm where}\hspace{1cm} \zeta^{-1} & \equiv & \sqrt{\frac{1}{2}\left(1 + \cos \theta \cos \frac{\phi}{2} \right)} \end{eqnarray} @@ -984,7 +1131,24 @@ \begin{eqnarray} -\phi & = & 2 {\rm arg} (2z^2 - 1, xz \pi/360^\circ) \\ -\theta & = & \arcsin (yz\pi/180^\circ) \\ -{\rm where}\hspace{1cm} z & \equiv & \sqrt{1 - (x\pi/720^\circ)^2 - (y\pi/360^\circ)^2} +\phi & = & 2 {\rm \arctan} (2z^2 - 1, x z) \\ +\theta & = & \arcsin (yz) \\ +{\rm where}\hspace{1cm} z & \equiv & \sqrt{1 - (x/2)^2 - y^2} +\end{eqnarray} + +\subparagraph{Parabolic} + +The Parabolic projection (``PAR'') is a pseudocylindrical projection, and is defined: + +\begin{eqnarray} +x & = & \phi \left( 2 \cos \frac{2 \theta}{3} - 1 \right) \\ +y & = & \pi \sin \frac{\theta}{3} \\ +\end{eqnarray} + +And in reverse: + +\begin{eqnarray} +\theta & = & 3 \sin^{-1} \rho \\ +\phi & = & \frac{x}{1 - 4\rho^2} \\ +{\rm where}\hspace{1cm} \rho & \equiv & y/\pi \\ \end{eqnarray}