/usr/share/axiom-20170501/src/algebra/COORDSYS.spad is in axiom-source 20170501-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 | )abbrev package COORDSYS CoordinateSystems
++ Author: Jim Wen
++ Date Created: 12 March 1990
++ Date Last Updated: 19 June 1990, Clifton J. Williamson
++ 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) : SIG == CODE where
R : Join(Field,TranscendentalFunctionCategory,RadicalCategory)
Pt ==> Point R
SIG ==> 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))}.
CODE ==> 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 ==
x+->
pt := copy(x)
u := elt(x,1); v := elt(x,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 ==
x+->
pt := copy(x)
xi := elt(x,1); eta := elt(x,2); phi := elt(x,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 ==
x+->
pt := copy(x)
xi := elt(x,1); eta := elt(x,2); phi := elt(x,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 ==
x+->
pt := copy(x)
u := elt(x,1); v := elt(x,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 ==
x+->
pt := copy(x)
u := elt(x,1); v := elt(x,2); phi := elt(x,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) ==
x+->
pt := copy(x)
lambda := elt(x,1); mu := elt(x,2); nu := elt(x,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
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