# 21.4 images5.input¶

The parameterization of the Etruscan Venus is due to George Frances. Etruscan Venus

venus(a,r,steps) ==
surf := (u:DFLOAT, v:DFLOAT): Point DFLOAT +->
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)                                  The Etruscan Venus

The Figure-8 Klein Bottle Klein bottle parameterization is from Differential Geometry and Computer Graphics by Thomas Banchoff, in Perspectives in Mathematics, Anniversary of Oberwolfasch 1984, Birkh\”{a}user-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@DFLOAT)
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,             Figure-8 Klein bottle
var2Steps==50,title==”Figure Eight Klein Bottle”)

The next two images are examples of generalized tubes.

)read ntube


rotateBy(p, theta) ==                                      Rotate a point p by

c := cos(theta)                                          θ around the origin
s := sin(theta) point [p.1*c - p.2*s, p.1*s + p.2*c]
bcircle t ==                                               A circle in three-space
point [3*cos t, 3*sin t, 0]

twist(u, t) ==                                             An ellipse that twists

theta := 4*t                                             around four times as

p := point [sin u, cos(u)/2]                             t revolves once
rotateBy(p, theta)
ntubeDrawOpt(bcircle, twist, 0..2*%pi, 0..2*%pi,           Twisted Torus
var1Steps == 70, var2Steps == 250)
twist2(u, t) ==                                            Create a twisting circle
theta := t p := point [sin u, cos(u)] rotateBy(p, theta)

cf(u,v) == sin(21*u)                                       Color function with 21 stripes

ntubeDrawOpt(bcircle, twist2, 0..2*%pi, 0..2*%pi,          Striped Torus
colorFunction == cf, var1Steps == 168, var2Steps == 126)