Curve of constant width
By Barbier's theorem, the body's perimeter is exactly π times its width, but its area depends on its shape, with the Reuleaux triangle having the smallest possible area for its width and the circle the largest.
Another application of curves of constant width is for coinage shapes, where regular Reuleaux polygons are a common choice.
The possibility that curves other than circles can have constant width makes it more complicated to check the roundness of an object.
Curves of constant width have been generalized in several ways to higher dimensions and to non-Euclidean geometry.
[2][3] Another equivalent way to define the width of a compact curve or of a convex set is by looking at its orthogonal projection onto a line.
So, a curve or a convex set has constant width when all of its orthogonal projections have the same length.
[3] Other curves of constant width can be smooth but non-circular, not even having any circular arcs in their boundary.
For instance, the zero set of the polynomial below forms a non-circular smooth algebraic curve of constant width:[4] Its degree, eight, is the minimum possible degree for a polynomial that defines a non-circular curve of constant width.
[5] Every regular polygon with an odd number of sides gives rise to a curve of constant width, a Reuleaux polygon, formed from circular arcs centered at its vertices that pass through the two vertices farthest from the center.
The line segment must be long enough to reach the cusp points of the curve, so that it can roll past each cusp to the next part of the curve, and its starting position should be carefully chosen so that at the end of the rolling process it is in the same position it started from.
The construction begins with a convex curved arc, whose endpoints are the intended width
[3] Victor Puiseux, a 19th-century French mathematician, found curves of constant width containing elliptical arcs[12] that can be constructed in this way from a semi-ellipse.
However, every curve of constant width can be enclosed by at least one regular hexagon with opposite sides on parallel supporting lines.
However, there exist other self-parallel curves, such as the infinite spiral formed by the involute of a circle, that do not have constant width.
[17][18] By the isoperimetric inequality and Barbier's theorem, the circle has the maximum area of any curve of given constant width.
The Blaschke–Lebesgue theorem says that the Reuleaux triangle has the least area of any convex curve of given constant width.
[22] A vertex of a smooth curve is a point where its curvature is a local maximum or minimum; for a circular arc, all points are vertices, but non-circular curves may have a finite discrete set of vertices.
This stands in contrast to the four-vertex theorem, according to which every simple closed smooth curve in the plane has at least four vertices.
[13][15] Because of the ability of curves of constant width to roll between parallel lines, any cylinder with a curve of constant width as its cross-section can act as a "roller", supporting a level plane and keeping it flat as it rolls along any level surface.
However, the center of the roller moves up and down as it rolls, so this construction would not work for wheels in this shape attached to fixed axles.
[2][6] Overlooking this fact may have played a role in the Space Shuttle Challenger disaster, as the roundness of sections of the rocket in that launch was tested only by measuring widths, and off-round shapes may cause unusually high stresses that could have been one of the factors causing the disaster.
[2][13] The curves of constant width may also be generalized to the bodies of constant brightness, three-dimensional shapes whose two-dimensional projections all have equal area; these shapes obey a generalization of Barbier's theorem.
