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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra coordsys.spad}
+\author{Jim Wen, Clifton J. Williamson}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{package COORDSYS CoordinateSystems}
+<<package COORDSYS CoordinateSystems>>=
+)abbrev package COORDSYS CoordinateSystems
+++ Author: Jim Wen
+++ Date Created: 12 March 1990
+++ Date Last Updated: 19 June 1990, Clifton J. Williamson
+++ Basic Operations: cartesian, polar, cylindrical, spherical, parabolic, elliptic, 
+++ parabolicCylindrical, paraboloidal, ellipticCylindrical, prolateSpheroidal,
+++ oblateSpheroidal, bipolar, bipolarCylindrical, toroidal, conical
+++ Related Constructors:
+++ Also See:
+++ AMS Classifications:
+++ Keywords:
+++ References:
+++ Description: CoordinateSystems provides coordinate transformation functions 
+++ for plotting.  Functions in this package return conversion functions 
+++ which take points expressed in other coordinate systems and return points 
+++ with the corresponding Cartesian coordinates.
+ 
+CoordinateSystems(R): Exports == Implementation where
+
+  R : Join(Field,TranscendentalFunctionCategory,RadicalCategory)
+  Pt ==> Point R
+
+  Exports ==> with
+    cartesian : Pt -> Pt
+      ++ cartesian(pt) returns the Cartesian coordinates of point pt.
+    polar: Pt -> Pt
+      ++ polar(pt) transforms pt from polar coordinates to Cartesian 
+      ++ coordinates: the function produced will map the point \spad{(r,theta)}
+      ++ to \spad{x = r * cos(theta)} , \spad{y = r * sin(theta)}.
+    cylindrical: Pt -> Pt
+      ++ cylindrical(pt) transforms pt from polar coordinates to Cartesian 
+      ++ coordinates: the function produced will map the point \spad{(r,theta,z)}
+      ++ to \spad{x = r * cos(theta)}, \spad{y = r * sin(theta)}, \spad{z}.
+    spherical: Pt -> Pt
+      ++ spherical(pt) transforms pt from spherical coordinates to Cartesian 
+      ++ coordinates: the function produced will map the point \spad{(r,theta,phi)}
+      ++ to \spad{x = r*sin(phi)*cos(theta)}, \spad{y = r*sin(phi)*sin(theta)},
+      ++ \spad{z = r*cos(phi)}.
+    parabolic: Pt -> Pt
+      ++ parabolic(pt) transforms pt from parabolic coordinates to Cartesian 
+      ++ coordinates: the function produced will map the point \spad{(u,v)} to
+      ++ \spad{x = 1/2*(u**2 - v**2)}, \spad{y = u*v}.
+    parabolicCylindrical: Pt -> Pt
+      ++ parabolicCylindrical(pt) transforms pt from parabolic cylindrical 
+      ++ coordinates to Cartesian coordinates: the function produced will 
+      ++ map the point \spad{(u,v,z)} to \spad{x = 1/2*(u**2 - v**2)}, 
+      ++ \spad{y = u*v}, \spad{z}.
+    paraboloidal: Pt -> Pt
+      ++ paraboloidal(pt) transforms pt from paraboloidal coordinates to 
+      ++ Cartesian coordinates: the function produced will map the point 
+      ++ \spad{(u,v,phi)} to \spad{x = u*v*cos(phi)}, \spad{y = u*v*sin(phi)},
+      ++ \spad{z = 1/2 * (u**2 - v**2)}.
+    elliptic: R -> (Pt -> Pt)
+      ++ elliptic(a) transforms from elliptic coordinates to Cartesian 
+      ++ coordinates: \spad{elliptic(a)} is a function which will map the 
+      ++ point \spad{(u,v)} to \spad{x = a*cosh(u)*cos(v)}, \spad{y = a*sinh(u)*sin(v)}.
+    ellipticCylindrical: R -> (Pt -> Pt)
+      ++ ellipticCylindrical(a) transforms from elliptic cylindrical coordinates 
+      ++ to Cartesian coordinates: \spad{ellipticCylindrical(a)} is a function
+      ++ which will map the point \spad{(u,v,z)} to \spad{x = a*cosh(u)*cos(v)},
+      ++ \spad{y = a*sinh(u)*sin(v)}, \spad{z}.
+    prolateSpheroidal: R -> (Pt -> Pt)
+      ++ prolateSpheroidal(a) transforms from prolate spheroidal coordinates to 
+      ++ Cartesian coordinates: \spad{prolateSpheroidal(a)} is a function 
+      ++ which will map the point \spad{(xi,eta,phi)} to 
+      ++ \spad{x = a*sinh(xi)*sin(eta)*cos(phi)}, \spad{y = a*sinh(xi)*sin(eta)*sin(phi)}, 
+      ++ \spad{z = a*cosh(xi)*cos(eta)}.
+    oblateSpheroidal: R -> (Pt -> Pt)
+      ++ oblateSpheroidal(a) transforms from oblate spheroidal coordinates to 
+      ++ Cartesian coordinates: \spad{oblateSpheroidal(a)} is a function which
+      ++ will map the point \spad{(xi,eta,phi)} to \spad{x = a*sinh(xi)*sin(eta)*cos(phi)},
+      ++ \spad{y = a*sinh(xi)*sin(eta)*sin(phi)}, \spad{z = a*cosh(xi)*cos(eta)}.
+    bipolar: R -> (Pt -> Pt)
+      ++ bipolar(a) transforms from bipolar coordinates to Cartesian coordinates:
+      ++ \spad{bipolar(a)} is a function which will map the point \spad{(u,v)} to
+      ++ \spad{x = a*sinh(v)/(cosh(v)-cos(u))}, \spad{y = a*sin(u)/(cosh(v)-cos(u))}.
+    bipolarCylindrical: R -> (Pt -> Pt)
+      ++ bipolarCylindrical(a) transforms from bipolar cylindrical coordinates 
+      ++ to Cartesian coordinates: \spad{bipolarCylindrical(a)} is a function which 
+      ++ will map the point \spad{(u,v,z)} to \spad{x = a*sinh(v)/(cosh(v)-cos(u))},
+      ++ \spad{y = a*sin(u)/(cosh(v)-cos(u))}, \spad{z}.
+    toroidal: R -> (Pt -> Pt)
+      ++ toroidal(a) transforms from toroidal coordinates to Cartesian 
+      ++ coordinates: \spad{toroidal(a)} is a function which will map the point 
+      ++ \spad{(u,v,phi)} to \spad{x = a*sinh(v)*cos(phi)/(cosh(v)-cos(u))},
+      ++ \spad{y = a*sinh(v)*sin(phi)/(cosh(v)-cos(u))}, \spad{z = a*sin(u)/(cosh(v)-cos(u))}.
+    conical: (R,R) -> (Pt -> Pt)
+      ++ conical(a,b) transforms from conical coordinates to Cartesian coordinates:
+      ++ \spad{conical(a,b)} is a function which will map the point \spad{(lambda,mu,nu)} to
+      ++ \spad{x = lambda*mu*nu/(a*b)},
+      ++ \spad{y = lambda/a*sqrt((mu**2-a**2)*(nu**2-a**2)/(a**2-b**2))},
+      ++ \spad{z = lambda/b*sqrt((mu**2-b**2)*(nu**2-b**2)/(b**2-a**2))}.
+
+  Implementation ==> add
+
+    cartesian pt ==
+      -- we just want to interpret the cartesian coordinates
+      -- from the first N elements of the point - so the
+      -- identity function will do
+      pt
+
+    polar pt0 ==
+      pt := copy pt0
+      r := elt(pt0,1); theta := elt(pt0,2)
+      pt.1 := r * cos(theta); pt.2 := r * sin(theta)
+      pt
+
+    cylindrical pt0 == polar pt0 
+    -- apply polar transformation to first 2 coordinates
+
+    spherical pt0 ==
+      pt := copy pt0
+      r := elt(pt0,1); theta := elt(pt0,2); phi := elt(pt0,3)
+      pt.1 := r * sin(phi) * cos(theta); pt.2 := r * sin(phi) * sin(theta)
+      pt.3 := r * cos(phi)
+      pt
+
+    parabolic pt0 ==
+      pt := copy pt0
+      u := elt(pt0,1); v := elt(pt0,2)
+      pt.1 := (u*u - v*v)/(2::R) ; pt.2 := u*v
+      pt
+
+    parabolicCylindrical pt0 == parabolic pt0
+    -- apply parabolic transformation to first 2 coordinates
+
+    paraboloidal pt0 ==
+      pt := copy pt0
+      u := elt(pt0,1); v := elt(pt0,2); phi := elt(pt0,3)
+      pt.1 := u*v*cos(phi); pt.2 := u*v*sin(phi); pt.3 := (u*u - v*v)/(2::R)
+      pt
+
+    elliptic a ==
+      pt := copy(#1)
+      u := elt(#1,1); v := elt(#1,2)
+      pt.1 := a*cosh(u)*cos(v); pt.2 := a*sinh(u)*sin(v)
+      pt
+
+    ellipticCylindrical a == elliptic a
+    -- apply elliptic transformation to first 2 coordinates
+
+    prolateSpheroidal a ==
+      pt := copy(#1)
+      xi := elt(#1,1); eta := elt(#1,2); phi := elt(#1,3)
+      pt.1 := a*sinh(xi)*sin(eta)*cos(phi)
+      pt.2 := a*sinh(xi)*sin(eta)*sin(phi)
+      pt.3 := a*cosh(xi)*cos(eta)
+      pt
+
+    oblateSpheroidal a ==
+      pt := copy(#1)
+      xi := elt(#1,1); eta := elt(#1,2); phi := elt(#1,3)
+      pt.1 := a*sinh(xi)*sin(eta)*cos(phi)
+      pt.2 := a*cosh(xi)*cos(eta)*sin(phi)
+      pt.3 := a*sinh(xi)*sin(eta)
+      pt
+
+    bipolar a ==
+      pt := copy(#1)
+      u := elt(#1,1); v := elt(#1,2)
+      pt.1 := a*sinh(v)/(cosh(v)-cos(u))
+      pt.2 := a*sin(u)/(cosh(v)-cos(u))
+      pt
+
+    bipolarCylindrical a == bipolar a
+    -- apply bipolar transformation to first 2 coordinates
+
+    toroidal a ==
+      pt := copy(#1)
+      u := elt(#1,1); v := elt(#1,2); phi := elt(#1,3)
+      pt.1 := a*sinh(v)*cos(phi)/(cosh(v)-cos(u))
+      pt.2 := a*sinh(v)*sin(phi)/(cosh(v)-cos(u))
+      pt.3 := a*sin(u)/(cosh(v)-cos(u))
+      pt
+
+    conical(a,b) ==
+      pt := copy(#1)
+      lambda := elt(#1,1); mu := elt(#1,2); nu := elt(#1,3)
+      pt.1 := lambda*mu*nu/(a*b)
+      pt.2 := lambda/a*sqrt((mu**2-a**2)*(nu**2-a**2)/(a**2-b**2))
+      pt.3 := lambda/b*sqrt((mu**2-b**2)*(nu**2-b**2)/(b**2-a**2))
+      pt
+
+@
+\section{License}
+<<license>>=
+--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
+--All rights reserved.
+--
+--Redistribution and use in source and binary forms, with or without
+--modification, are permitted provided that the following conditions are
+--met:
+--
+--    - Redistributions of source code must retain the above copyright
+--      notice, this list of conditions and the following disclaimer.
+--
+--    - Redistributions in binary form must reproduce the above copyright
+--      notice, this list of conditions and the following disclaimer in
+--      the documentation and/or other materials provided with the
+--      distribution.
+--
+--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
+--      names of its contributors may be used to endorse or promote products
+--      derived from this software without specific prior written permission.
+--
+--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
+--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
+--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
+--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
+--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
+--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+@
+<<*>>=
+<<license>>
+
+<<package COORDSYS CoordinateSystems>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}