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
|
\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input conformal.input}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{License}
<<license>>=
--Copyright The Numerical Algorithms Group Limited 1994.
@
<<*>>=
<<license>>
-- Drawing conformal maps.
-- The functions in this file draw conformal maps both on the
-- complex plane and on the Riemann sphere.
-- Compile, don't interpret functions.
)set fun comp on
C := Complex DoubleFloat -- Complex Numbers
S := Segment DoubleFloat -- Draw ranges
R3 := POINT DoubleFloat -- points in 3-space
-- conformalDraw(f, rRange, tRange, rSteps, tSteps, coord)
-- draws the image of the coordinate grid under f in the complex plane.
-- The grid may be given in either polar or cartesian coordinates.
-- parameter descriptions:
-- f: the function to draw
-- rRange: the range of the radius (in polar) or real (in cartesian)
-- tRange: the range of theta (in polar) or imaginary (in cartesian)
-- tSteps, rSteps: the number of intervals in each direction
-- coord: the coordinate system to use. Either "polar" or "cartesian"
conformalDraw: (C -> C, S, S, PI, PI, String) -> VIEW3D
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")
-- riemannConformalDraw(f, rRange, tRange, rSteps, tSteps, coord)
-- draws the image of the coordinate grid under f on the Riemann sphere.
-- The grid may given in either polar or cartesian coordinates.
-- parameter descriptions:
-- f: the function to draw
-- rRange: the range of the radius(in polar) or real (in cartesian)
-- tRange: the range of theta (in polar) or imaginary (in cartesian)
-- tSteps, rSteps: the number of intervals in each direction
-- coord: the coordinate system to use. either "polar" or "cartesian"
riemannConformalDraw: (C -> C, S, S, PI, PI, String) -> VIEW3D
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")
-- Plot the coordinate grid using adaptive plotting for the coordinate
-- lines, and drawing tubes around the lines.
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()
-- map a point in the complex plane to the Riemann sphere.
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]
-- convert cartesian coordinates to cartesian form complex
cartesian2Complex(r:DoubleFloat, i:DoubleFloat):C == complex(r, i)
-- convert polar coordinates to cartesian form complex
polar2Complex(r:DoubleFloat, th:DoubleFloat):C == complex(r*cos(th), r*sin(th))
-- convert a complex function into a mapping from (DoubleFloat,DoubleFloat) to R3 in the
-- complex plane.
makeConformalMap(f, transformC) ==
(u:DoubleFloat,v:DoubleFloat):R3 +->
z := f transformC(u, v)
point [real z, imag z, 0.0@DoubleFloat]
-- convert a complex function into a mapping from (DoubleFloat,DoubleFloat) to R3 on the
-- Riemann sphere.
makeRiemannConformalMap(f, transformC) ==
(u:DoubleFloat, v:DoubleFloat):R3 +-> riemannTransform f transformC(u, v)
-- draw a picture of the mapping of the complex plane to the Riemann sphere.
riemannSphereDraw: (S, S, PI, PI, String) -> VIEW3D
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")
-- draw the lines which connect the points in the complex plane to
-- the north pole of the Riemann sphere.
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()
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]
-- create a line functions which goeas from p1 to p2 as its paramter
-- goes from 0 to 1.
lineFromTo(p1, p2) ==
d := p2 - p1
(t:DoubleFloat):Point DoubleFloat +-> p1 + t*d
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
