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Curves and Surfaces of Constant Width

Curves of constant width are useful for noncircular coins. As the number of sides increases, these curves quickly become more like disks and less like Reuleaux triangles. The curves here are defined using a simple support function: for an odd integer, .
To preserve convexity, is needed. Support functions have a central role in the definition of sets of constant width; for example, in the Eggleston (1952) proof of the Blaschke–Lebesgue (1914) theorem, the Reuleaux triangle is the planar set of constant width of minimal area.

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Rotating a curve of constant width about an axis of symmetry creates a surface of constant width [1, p. 196], but there are other kinds of surfaces of constant width. There are curves of constant width without an axis of symmetry [2].
References
[1] J. Bryant and C. Sangwin, How Round Is Your Circle?, Princeton, NJ: Princeton University Press, 2008 pp. 188–226.
[2] A. Bogomolny. "Star Construction of Shapes of Constant Width." (Apr 16, 2013) www.cut-the-knot.org/Curriculum/Geometry/CWStar.shtml.
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