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\documentclass{article}
\usepackage{open-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}
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