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
|
\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input images1a.input}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{License}
<<license>>=
--Copyright The Numerical Algorithms Group Limited 1994-1996.
@
<<*>>=
<<license>>
-- Create a (p,q) torus-knot with radius r around the curve.
-- The formula was derived by Larry Lambe.
-- To produce a trefoil knot:
-- torusKnot(2, 3, 0.5, 10, 200)
-- compile, don't interpret functions
)set function compile on
-- Generalized tubes.
-- These functions draw a 2-d curve in the normal
-- planes around a 3-d curve. The computations are all done
-- numerically in machine-precision floating point for efficiency.
R3 := Point DoubleFloat -- Points in 3-Space
R2 := Point DoubleFloat -- Points in 2-Space
S := Segment Float -- Draw ranges
ThreeCurve := DoubleFloat -> R3 -- type of a space curve function
TwoCurve := (DoubleFloat, DoubleFloat) -> R2 -- type of a plane curve function
Surface := (DoubleFloat, DoubleFloat) -> R3 -- type of a parameterized surface function
-- Frenet frames define a coordinate system around a point on a space curve
FrenetFrame := Record(value: R3, tagent: R3, normal: R3, binormal: R3)
-- Holds current Frenet frame for a point on a curve
frame: FrenetFrame
-- compile, don't interpret functions
)set fun compile on
-- Draw a generalized tube.
-- ntubeDraw(spaceCurve, planeCurve, u0..u1, t0..t1)
-- draws planeCurve int the normal planes of spaceCurve. u0..u1 specifies
-- the paramter range of the planeCurve and t0..t1 specifies the parameter
-- range of the spaceCurve. Additionally the plane curve function takes
-- as a second parameter the current parameter of the spaceCurve. This
-- allows the plane curve to evolve as it goes around the space curve.
-- see "page5.input" for an example of this.
ntubeDraw: (ThreeCurve, TwoCurve, S, S) -> VIEW3D
ntubeDraw(spaceCurve, planeCurve, uRange, tRange) ==
ntubeDrawOpt(spaceCurve, planeCurve, uRange, tRange, []$List DROPT)
-- ntuberDrawOpt is the same as ntuberDraw, but takes optional
-- parameters which it passes to the draw command.
ntubeDrawOpt: (ThreeCurve, TwoCurve, S, S, List DROPT) -> VIEW3D
ntubeDrawOpt(spaceCurve, planeCurve, uRange, tRange, l) ==
delT:DoubleFloat := (hi(tRange) - lo(tRange))/10000
oldT:DoubleFloat := lo(tRange) - 1
fun := ngeneralTube(spaceCurve, planeCurve, delT, oldT)
draw(fun, uRange, tRange, l)
-- nfrenetFrame(c, t, delT) numerically computes the Frenet Frame
-- about the curve c at t. delT is a small number used to
-- compute derivatives.
nfrenetFrame(c, t, delT) ==
f0 := c(t)
f1 := c(t+delT)
t0 := f1 - f0 -- the tangent
n0 := f1 + f0
b := cross(t0, n0) -- the binormal
n := cross(b,t0) -- the normal
ln := length n
lb := length b
ln = 0 or lb = 0 => error "Frenet Frame not well defined"
n := (1/ln)*n -- make into unit length vectors
b := (1/lb)*b
[f0, t0, n, b]$FrenetFrame
-- nGeneralTube(spaceCurve, planeCurve, delT, oltT)
-- creates a function which can be passed to the system draw command.
-- The function is a parameterized surface for the general tube
-- around the spaceCurve. delT is a small number used to compute
-- derivatives, and oldT is used to hold the current value of the
-- t parameter for the spaceCurve. This is an efficiency measure
-- to ensure that frames are only computed once for every value of t.
ngeneralTube: (ThreeCurve, TwoCurve, DoubleFloat, DoubleFloat) -> Surface
ngeneralTube(spaceCurve, planeCurve, delT, oldT) ==
free frame
(v:DoubleFloat, t: DoubleFloat): R3 +->
if (t ~= oldT) then
frame := nfrenetFrame(spaceCurve, t, delT)
oldT := t
p := planeCurve(v, t)
frame.value + p.1*frame.normal + p.2*frame.binormal
-- Create a (p,q) torus-knot with radius r around the curve.
-- The formula was derived by Larry Lambe.
-- To produce a trefoil knot:
-- torusKnot(2, 3, 0.5)
torusKnot(p:DFLOAT, q:DFLOAT, r:DFLOAT, uSteps:PI, tSteps:PI):VIEW3D ==
-- equation for the torus knot
knot := (t:DFLOAT):Point DFLOAT +->
fac := 4/(2.2@DFLOAT-sin(q*t))
fac * point [cos(p*t), sin(p*t), cos(q*t)]
-- equation for the cross section of the tube
circle := (u:DFLOAT, t:DFLOAT):Point DFLOAT +->
r * point [cos u, sin u]
-- draw the tube around the knot
ntubeDrawOpt(knot, circle, 0..2*%pi, 0..2*%pi, var1Steps == uSteps,
var2Steps == tSteps)
-- draw a (15,17) torus knot
torusKnot(15,17, 0.1, 6, 700)
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
