Editing Curve of constant width (section)

{{short description|Shape with width independent of orientation}}
{{good article}}
[[File:Reuleaux supporting lines.svg|thumb|Measuring the width of a [[Reuleaux triangle]] as the distance between parallel [[supporting line]]s. Because this distance does not depend on the direction of the lines, the Reuleaux triangle is a curve of constant width.]]
In [[geometry]], a '''curve of constant width''' is a [[simple closed curve]] in the [[plane (geometry)|plane]] whose width (the distance between parallel [[supporting line]]s) is the same in all directions. The shape bounded by a curve of constant width is a '''body of constant width''' or an '''orbiform''', the name given to these shapes by [[Leonhard Euler]].{{r|euler}} Standard examples are the [[circle]] and the [[Reuleaux triangle]]. These curves can also be constructed using circular arcs centered at crossings of an [[arrangement of lines]], as the [[involute]]s of certain curves, or by intersecting circles centered on a partial curve.

Every body of constant width is a [[convex set]], its boundary crossed at most twice by any line, and if the line crosses perpendicularly it does so at both crossings, separated by the width. By [[Barbier's theorem]], the body's perimeter is exactly [[pi|{{pi}}]] 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. Every superset of a body of constant width includes pairs of points that are farther apart than the width, and every curve of constant width includes at least six points of extreme curvature. Although the Reuleaux triangle is not smooth, curves of constant width can always be approximated arbitrarily closely by smooth curves of the same constant width.

Cylinders with constant-width cross-section can be used as rollers to support a level surface. Another application of curves of constant width is for [[coinage shapes]], where regular [[Reuleaux polygon]]s are a common choice. The possibility that curves other than circles can have constant width makes it more complicated to check the [[Roundness (object)|roundness of an object]].

Curves of constant width have been generalized in several ways to higher dimensions and to [[non-Euclidean geometry]].