Historical Sketch: A constant width curve is a planar convex oval with the property that the distance between two parallel tangents to the curve is constant. Visualize a circle inscribed in square with the circle rolling, or rotating in the square. The diameter of the circle is the same as the width of the square. The width of a closed convex curve is defined to be the distance between the parallel lines bounding it. The parallel lines of the square are sometimes called "supporting lines." Please note, the inscribed asteroid does not fit the definition. Some background is helpful. Unlike many plane curves, the constant width and Reuleaux polygon investigations are rooted in machine design and engineering. Moreover, compared to the history of most plane curves, this work is relatively young. Franz Reuleaux (1829  1905) recognized that simple plane
curves of constant width might be constructed from regular polygons
with an odd number of sides. Thus, triangles and pentagons are
frequently constructed using a corresponding number of intersecting
arcs. In engineering, Felix Heinrch Wankel (19021988) designed a
rotor engine which has the shape of a Reuleaux triangle inscribed in a
chamber, rather than the usual piston, cylinder, and mechanical
valves. The rotor engine, now found in Mazda automobiles has 40%
fewer parts and thus far less weight. Within the Wankel rotor,
three chambers are formed by the sides of the rotor and the wall of the
housing. The shape, size, and position of the chambers are constantly
altered by the rotation of the rotor, i.e., the Reuleaux triangle or
deltoid. _
