Curve of constant width: Difference between revisions

Revision as of 00:48, 11 January 2021

Revision as of 00:49, 11 January 2021

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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, forsince 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}}