[back]  3.8. Little Bottle 
[next] 
The socalled Klein bottle [ 6,9,10,11,19 ] was named after the mathematician Felix Klein. It is a threedimensional extension of the Mbius strip. Figuratively presented, the Klein bottle is a tube that has been twisted and folded over the fourth dimension in such a way that an object traveling through it ends its journey the wrong way round.
The Klein bottle is represented by the following equations. To keep the formulas a little clearer, we use the constant r.
r = 4 (1  cos(u)/2) 
330 
To represent the Klein Bottle, different equations are required for parts of the domain of u.
0 <= u < pi 
331 

x = a cos(u) (1 + sin(u)) + r cos(u) cos(v) 
332 

y = b sin(u) + r sin(u) cos(v) 
333 

z = r sin(v) 
334 
pi < u <= 2pi 
335 

x = a cos(u) (1 + sin(u)) + r cos(v + pi) 
336 

y = bsin(u) 
337 

z = r sin(v) 
338 
The constants a and b determine the appearance of the figure.
To represent the area, the two parameters u and v must have the following values (definition range).
u is an element from the set of numbers [0, 2 pi] 

v is an element of the number set [0, 2 pi] 
Since the Klein Bottle is a closed figure, the definition range must be adhered to exactly, so it cannot be changed with the plugin.
The plugin creates an optimized mesh without duplicate points and unconnected polygons.
[back]  [Table of contents]  [next] 