figure 8 Klien bottle

The Klein bottle is a closed nonorientable surface that has no inside or outside.

Houdini Math Geometry.

Date Created:Tuesday January 16th, 2007 12:18 AM
Date Modified:Saturday August 02nd, 2008 12:30 AM

figure 8 Klien bottle -

figure 8 Klien bottle -

Figure 8 Klein bottle

The "figure-8" form of the Klein bottle is obtained by rotating a figure eight about an axis while placing a twist in it, and is given by parametric equations
x(u,v) = [a+cos(1/2u)sin(v)-sin(1/2u)sin(2v)]cos(u)
y(u,v) = [a+cos(1/2u)sin(v)-sin(1/2u)sin(2v)]sin(u)
z(u,.v) = sin(1/2u)sin(v)+cos(1/2u)sin(2v)
for u in [0,2pi), v in [0,2pi), and a>2

In Houdini we can visualize with a $ROWS*$COLS grid
Create the variables u and v then enter the following the equations into the x,y,z pos parameters of a point SOP.
x pos = (2+cos($U/2)*sin($V)-sin($U/2)*sin(2*$V))*cos($U)
y pos = (2+cos($U/2)*sin($V)-sin($U/2)*sin(2*$V))*sin($U)
z pos = sin($U/2)*sin($V)+cos($U/2)*sin(2*$V)
$U = frac($PT/$ROWS)*360 *$COLS/($COLS-1)
$V = floor($PT/$ROWS)/($COLS-1)*360

Download: figure_8_klein_bottle.hipnc 149 KB

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figure 8 Klien bottle by Dan Lynch
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