Editing Curve of constant width (section)

==Properties==
[[File:Reuleaux triangle Animation.gif|thumb|The Reuleaux triangle rolling within a square while at all times touching all four sides]]
A curve of constant width can rotate between two parallel lines separated by its width, while at all times touching those lines, which act as supporting lines for the rotated curve. In the same way, a curve of constant width can rotate within a rhombus or square, whose pairs of opposite sides are separated by the width and lie on parallel support lines.{{r|gardner|bs|rt}} Not every curve of constant width can rotate within a regular [[hexagon]] in the same way, because its supporting lines may form different irregular hexagons for different rotations rather than always forming a regular one. However, every curve of constant width can be enclosed by at least one regular hexagon with opposite sides on parallel supporting lines.{{r|chakerian}}

A curve has constant width if and only if, for every pair of parallel supporting lines, it touches those two lines at points whose distance equals the separation between the lines. In particular, this implies that it can only touch each supporting line at a single point. Equivalently, every line that crosses the curve perpendicularly crosses it at exactly two points of distance equal to the width. Therefore, a curve of constant width must be convex, since every non-convex simple closed curve has a supporting line that touches it at two or more points.{{r|rt|robertson}} Curves of constant width are examples of self-parallel or auto-parallel curves, curves traced by both endpoints of a line segment that moves in such a way that both endpoints move perpendicularly to the line segment. 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.{{r|mathcurve}}

[[Barbier's theorem]] asserts that the [[perimeter]] of any curve of constant width is equal to the width multiplied by <math>\pi</math>. As a special case, this formula agrees with the standard formula <math>\pi d</math> for the perimeter of a circle given its diameter.{{r|lay|barbier}} 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.{{r|gruber}} Every proper superset of a body of constant width has strictly greater diameter, and every Euclidean set with this property is a body of constant width. In particular, it is not possible for one body of constant width to be a subset of a different body with the same constant width.{{r|eggleston|jessen}} Every curve of constant width can be approximated arbitrarily closely by a piecewise circular curve or by an [[analytic curve]] of the same constant width.{{r|wegner77}}

A [[vertex (curve)|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. For a curve that is not smooth, the points where it is not smooth can also be considered as vertices, of infinite curvature. For a curve of constant width, each vertex of locally minimum curvature is paired with a vertex of locally maximum curvature, opposite it on a diameter of the curve, and there must be at least six 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. Some curves, such as ellipses, have exactly four vertices, but this is not possible for a curve of constant width.{{r|martinez|ctb}} Because local minima of curvature are opposite local maxima of curvature, the only curves of constant width with [[central symmetry]] are the circles, for which the curvature is the same at all points.{{r|mmo}} For every curve of constant width, the [[Smallest-circle problem|minimum enclosing circle]] of the curve and the largest circle that it contains are concentric, and the average of their diameters is the width of the curve. These two circles together touch the curve in at least three pairs of opposite points, but these points are not necessarily vertices.{{r|mmo}}

A convex body has constant width if and only if the Minkowski sum of the body and its 180° rotation is a circular disk; if so, the width of the body is the radius of the disk.{{r|mmo|chakerian}}