[back]  3.8. Little Bottle 
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The socalled Klein bottle [ 6,9,10,11,19 ] was named after the mathematician Felix Klein. It is a threedimensional extension of the Möbius 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) 
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To represent the Klein Bottle, different equations are required for parts of the domain of u.
0 <= u < pi 
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x = a cos(u) (1 + sin(u)) + r cos(u) cos(v) 
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y = b sin(u) + r sin(u) cos(v) 
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z = r sin(v) 
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pi < u <= 2pi 
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x = a cos(u) (1 + sin(u)) + r cos(v + pi) 
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y = bsin(u) 
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z = r sin(v) 
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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.
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