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[back] | 3.8. Little Bottle |
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The so-called Klein bottle [ 6,9,10,11,19 ] was named after the mathematician Felix Klein. It is a three-dimensional 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.
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r = 4 (1 - cos(u)/2) |
3-30 |
To represent the Klein Bottle, different equations are required for parts of the domain of u.
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0 <= u < pi |
3-31 |
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x = a cos(u) (1 + sin(u)) + r cos(u) cos(v) |
3-32 |
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y = b sin(u) + r sin(u) cos(v) |
3-33 |
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z = r sin(v) |
3-34 |
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pi < u <= 2pi |
3-35 |
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x = a cos(u) (1 + sin(u)) + r cos(v + pi) |
3-36 |
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y = bsin(u) |
3-37 |
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z = r sin(v) |
3-38 |
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).
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u is an element from the set of numbers [0, 2 pi] |
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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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