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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input images5a.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>>
-- Etruscan Venus
-- Parameterization by George Frances
venus(a,r,steps) ==
surf := (u:DoubleFloat, v:DoubleFloat): Point DoubleFloat +->
cv := cos(v)
sv := sin(v)
cu := cos(u)
su := sin(u)
x := r * cos(2*u) * cv + sv * cu
y := r * sin(2*u) * cv - sv * su
z := a * cv
point [x,y,z]
draw(surf, 0..%pi, -%pi..%pi, var1Steps==steps,var2Steps==steps,
title == "Etruscan Venus")
venus(5/2, 13/10, 50)
-- Figure Eight Klein Bottle
-- Parameterization from:
-- "Differential Geometry and Computer Graphics" by Thomas Banchoff
-- in Perspectives in Mathemtaics, Anneversry of Oberwolfasch 1984.
-- Beirkhauser-Verlag, Basel, pp 43-60.
klein(x,y) ==
cx := cos(x)
cy := cos(y)
sx := sin(x)
sy := sin(y)
sx2 := sin(x/2)
cx2 := cos(x/2)
sq2 := sqrt(2.0@DoubleFloat)
point [cx * (cx2 * (sq2 + cy) + (sx2 * sy * cy)), _
sx * (cx2 * (sq2 + cy) + (sx2 * sy * cy)), _
-sx2 * (sq2 + cy) + cx2 * sy * cy]
draw(klein, 0..4*%pi, 0..2*%pi, var1Steps==50, var2Steps==50, _
title=="Figure Eight Klein Bottle")
-- Twisted torus
-- Generalized tubes.
-- The functions in this file 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
-- rotate a 2-d point by theta round the origin
rotateBy(p, theta) ==
c := cos(theta)
s := sin(theta)
point [p.1*c - p.2*s, p.1*s + p.2*c]
-- a circle in 3-space
bcircle t ==
point [3*cos t, 3*sin t, 0]
-- an elipse which twists around 4 times as t revolves once.
twist(u, t) ==
theta := 4*t
p := point [sin u, cos(u)/2]
rotateBy(p, theta)
ntubeDrawOpt(bcircle, twist, 0..2*%pi, 0..2*%pi, _
var1Steps == 70, var2Steps == 250)
-- Striped torus
-- a twisting circle
twist2(u, t) ==
theta := t
p := point [sin u, cos(u)]
rotateBy(p, theta)
-- color function producing 21 stripes
cf(u,v) == sin(21*u)
ntubeDrawOpt(bcircle, twist2, 0..2*%pi, 0..2*%pi, _
colorFunction == cf, var1Steps == 168, var2Steps == 126)
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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