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## 3.9. Moebius band

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The Möbius strip [ 6,7,11 ] was named after the German mathematician and astronomer August Ferdinand Möbius (1790-1868). Anyone who is interested should make a Möbius strip themselves at least once. Simply take a long, narrow strip of paper and glue the ends (twisted by 180°) together.

Now this band has some strange properties. It only has one page, although it doesn't really look like it. If you don't believe it, you should try coloring one side of the ribbon or drawing a line along the ribbon on one side.

The band also has only one edge. If you don't believe it, you should try cutting the ribbon lengthways with a pair of scissors. It will not succeed, because a band with an edge remains a single band even after cutting. It gets even more amazing when we try to cut that ribbon again.

The Möbius strip is represented by the following equations.

 x = [R + s cos(t/2)] cos(t) 3-39 y = [R + s cos(t/2)] sin(t) 3-40 z = s sin(t/2) 3-41

The constant R determines the radius of the band. With the parameter s = 0 we get a circle with radius R and height z = 0.

The constant w determines the width of the band.

To represent the area, the two parameters s and t must have the following values ​​(definition range).

 s is an element from the number set [-w, w] t is an element from the number set [0, 2 pi]

Since the Möbius strip is a closed figure, the domain of definition for the parameter t must be observed exactly.
The plugin creates an optimized mesh without duplicate points and unconnected polygons.

Fig. 13