Crossed-Lines Construction of Shapes of Constant Width

In the star construction of the shapes of constant width, the boundary curve consists of circular arcs of two radii as in the case of the Reuleaux triangle. A Crossed-Lines method produces the curves that are still composed of circular arcs, but with the number of radii depending on the number of lines and their configuration. For three lines, as in the applet below, the number of different radii may be six.

There are four controls: three points of intersection of the three lines taken by two and an additional point on one of the lines. To obtain the curve, the point is rotated around the nearest vertex, to the next line. The latter meets the third line in another control point, which is now taken to be the next center of rotation and so on. The process is exactly the same as in one of Howard Eves' problems, where you can also find a proof of the fact that after six rotations the curve closes in and have the constant width.

In general, the method works with more than three lines and perhaps in the future I'll find time for a more general applet. As it is right now, the controls can be so moved as to produce a non-convex curve. The applet is not made to warn you that the resulting curve does not have constant width, we should be obvious enough.

References

1. M. Gardner, The Unexpected Hanging and Other Mathematical Diversions, The University of Chicago Press, 1991, pp. 216-217
2. H. Rademacher and O.Toeplitz, The Enjoyment of Mathematics, Dover Publications, 1990, p. 167.

Copyright © 1996-2008 Alexander Bogomolny