Curve of constant width: Difference between revisions

Revision as of 05:34, 9 January 2021

Revision as of 05:37, 9 January 2021

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==Examples==

[[File:A curve of constant width defined by 8th-degree polynomial.png|thumb|A curve of constant width defined by an 8th-degree polynomial]]

[[Circle]]s have constant width, equal to their [[diameter]]. On the other hand, squares do not: supporting lines parallel to two opposite sides of the square are closer together than supporting lines parallel to a diagonal. More generally, no [[polygon]] can have constant width. However, there are many non-circularother shapes of constant width. A standard example is the [[Reuleaux triangle]], the intersection of three circles, each centered where the other two circles cross.{{r|gardner}} Its boundary curve consists of three arcs of these circles, meeting at 120° angles, so it is not [[Smooth function#Smoothness of curves and surfaces|smooth]], and in fact these angles are the sharpest possible for any curve of constant width.{{r|rt}}

Other curves of constant width can be smooth but non-circular, not even having any circular arcs in their boundary.