/usr/share/axiom-20170501/input/conformal.input is in axiom-test 20170501-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 | )set break resume
)set fun comp on
)spool conformal.output
)set message test on
)set message auto off
)clear all
--S 1 of 18
C := Complex DoubleFloat -- Complex Numbers
--R
--R
--R (1) Complex(DoubleFloat)
--R Type: Domain
--E 1
--S 2 of 18
S := Segment DoubleFloat -- Draw ranges
--R
--R
--R (2) Segment(DoubleFloat)
--R Type: Domain
--E 2
--S 3 of 18
R3 := POINT DoubleFloat -- points in 3-space
--R
--R
--R (3) Point(DoubleFloat)
--R Type: Domain
--E 3
--S 4 of 18
conformalDraw: (C -> C, S, S, PI, PI, String) -> VIEW3D
--R
--R Type: Void
--E 4
--S 5 of 18
conformalDraw(f, rRange, tRange, rSteps, tSteps, coord) ==
transformC :=
coord = "polar" => polar2Complex
cartesian2Complex
cm := makeConformalMap(f, transformC)
sp := createThreeSpace()
adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps)
makeViewport3D(sp, "Conformal Map")
--R
--R Type: Void
--E 5
--S 6 of 18
riemannConformalDraw: (C -> C, S, S, PI, PI, String) -> VIEW3D
--R
--R Type: Void
--E 6
--S 7 of 18
riemannConformalDraw(f, rRange, tRange, rSteps, tSteps, coord) ==
transformC :=
coord = "polar" => polar2Complex
cartesian2Complex
sp := createThreeSpace()
cm := makeRiemannConformalMap(f, transformC)
adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps)
-- add an invisible point at the north pole for scaling
curve(sp, [point [0,0,2.0@DoubleFloat,0], point [0,0, 2.0@DoubleFloat,0]])
makeViewport3D(sp, "Conformal Map on the Riemann Sphere")
--R
--R Type: Void
--E 7
--S 8 of 18
adaptGrid(sp, f, uRange, vRange, uSteps, vSteps) ==
delU := (hi(uRange) - lo(uRange))/uSteps
delV := (hi(vRange) - lo(vRange))/vSteps
uSteps := uSteps + 1; vSteps := vSteps + 1
u := lo uRange
-- draw the coodinate lines in the v direction
for i in 1..uSteps repeat
-- create a curve 'c' which fixes the current value of 'u'
c := curryLeft(f,u)
cf := (t:DoubleFloat):DoubleFloat +-> 0
-- draw the 'v' coordinate line
makeObject(c, vRange::Segment Float, colorFunction == cf, space == sp, _
tubeRadius == 0.02, tubePoints == 6)
u := u + delU
v := lo vRange
-- draw the coodinate lines in the u direction
for i in 1..vSteps repeat
-- create a curve 'c' which fixes the current value of 'v'
c := curryRight(f,v)
cf := (t:DoubleFloat):DoubleFloat +-> 1
-- draw the 'u' coordinate line
makeObject(c, uRange::Segment Float, colorFunction == cf, space == sp, _
tubeRadius == 0.02, tubePoints == 6)
v := v + delV
void()
--R
--R Type: Void
--E 8
--S 9 of 18
riemannTransform(z) ==
r := sqrt norm z
cosTheta := (real z)/r
sinTheta := (imag z)/r
cp := 4*r/(4+r^2)
sp := sqrt(1-cp*cp)
if r>2 then sp := -sp
point [cosTheta*cp, sinTheta*cp, -sp + 1]
--R
--R Type: Void
--E 9
--S 10 of 18
cartesian2Complex(r:DoubleFloat, i:DoubleFloat):C == complex(r, i)
--R
--R Function declaration cartesian2Complex : (DoubleFloat,DoubleFloat)
--R -> Complex(DoubleFloat) has been added to workspace.
--R Type: Void
--E 10
--S 11 of 18
polar2Complex(r:DoubleFloat, th:DoubleFloat):C == complex(r*cos(th), r*sin(th))
--R
--R Function declaration polar2Complex : (DoubleFloat,DoubleFloat) ->
--R Complex(DoubleFloat) has been added to workspace.
--R Type: Void
--E 11
--S 12 of 18
makeConformalMap(f, transformC) ==
(u:DoubleFloat,v:DoubleFloat):R3 +->
z := f transformC(u, v)
point [real z, imag z, 0.0@DoubleFloat]
--R
--R Type: Void
--E 12
--S 13 of 18
makeRiemannConformalMap(f, transformC) ==
(u:DoubleFloat, v:DoubleFloat):R3 +-> riemannTransform f transformC(u, v)
--R
--R Type: Void
--E 13
--S 14 of 18
riemannSphereDraw: (S, S, PI, PI, String) -> VIEW3D
--R
--R Type: Void
--E 14
--S 15 of 18
riemannSphereDraw(rRange, tRange, rSteps, tSteps, coord) ==
transformC :=
coord = "polar" => polar2Complex
cartesian2Complex
grid := (u:DoubleFloat , v:DoubleFloat): R3 +->
z1 := transformC(u, v)
point [real z1, imag z1, 0]
sp := createThreeSpace()
adaptGrid(sp, grid, rRange, tRange, rSteps, tSteps)
connectingLines(sp, grid, rRange, tRange, rSteps, tSteps)
makeObject(riemannSphere, 0..2*%pi, 0..%pi, space == sp)
f := (z:C):C +-> z
cm := makeRiemannConformalMap(f, transformC)
adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps)
makeViewport3D(sp, "Riemann Sphere")
--R
--R Type: Void
--E 15
--S 16 of 18
connectingLines(sp, f, uRange, vRange, uSteps, vSteps) ==
delU := (hi(uRange) - lo(uRange))/uSteps
delV := (hi(vRange) - lo(vRange))/vSteps
uSteps := uSteps + 1; vSteps := vSteps + 1
u := lo uRange
-- for each grid point
for i in 1..uSteps repeat
v := lo vRange
for j in 1..vSteps repeat
p1 := f(u,v)
p2 := riemannTransform complex(p1.1, p1.2)
fun := lineFromTo(p1,p2)
cf := (t:DoubleFloat):DoubleFloat +-> 3
makeObject(fun, 0..1, space == sp, tubePoints == 4, tubeRadius == 0.01,
colorFunction == cf)
v := v + delV
u := u + delU
void()
--R
--R Type: Void
--E 16
--S 17 of 18
riemannSphere(u,v) ==
sv := sin(v)
0.99@DoubleFloat*(point [cos(u)*sv, sin(u)*sv, cos(v),0.0@DoubleFloat]) +
point [0.0@DoubleFloat, 0.0@DoubleFloat, 1.0@DoubleFloat, 4.0@DoubleFloat]
--R
--R Type: Void
--E 17
--S 18 of 18
lineFromTo(p1, p2) ==
d := p2 - p1
(t:DoubleFloat):Point DoubleFloat +-> p1 + t*d
--R
--R Type: Void
--E 18
)spool
)lisp (bye)
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