Curve of constant width: Difference between revisions

Revision as of 21:34, 10 January 2021

Revision as of 21:42, 10 January 2021

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[[Leonhard Euler]] constructed curves of constant width as the [[involute]]s of curves with an odd number of [[Cusp (singularity)|cusp singularities]], having only one [[tangent line]] in each direction (that is, [[Hedgehog (geometry)|projective hedgehogs]]). If the starting curve is smooth (except at the cusps), the resulting curve of constant width will also be smooth.{{r|euler|robertson}} An example of a starting curve with the correct properties for this construction is the [[deltoid curve]], and involutes of the deltoid form smooth curves of constant width, not containing any circular arcs.{{r|goldberg|burke}} The same construction can also be obtained by rolling a line segment along the same starting curve, without sliding it, until it returns to its starting position. All long-enough line segments have a starting position that returns in this way.{{r|lowry}}

Another construction chooses half of the curve of constant width, meeting certain requirements, and forms from it a body of constant width having the given curve as part of its boundary. The construction begins with a convex curved arc, whose endpoints are the intended width <math>w</math> apart. The two endpoints must touch parallel supporting lines at distance <math>w</math> from each other, and each supporting line betweenthat thesetouches twoanother lines (on the convex sidepoint of the curve)arc must be tangent at that point to a circle of radius <math>w</math> containing the entire arc. This requirement prevents the [[curvature]] of the arc from being less than that of the circle. The completed body of constant width is then the intersection of the interiors of an infinite family of circles, of two types: the ones tangent to the supporting lines, and more circles of the same radius centered at each point of the given arc. This construction is universal: all curves of constant width may be constructed in this way.{{r|rt}} [[Victor Puiseux]], a 19th-century French mathematician, found curves of constant width containing elliptical arcs{{r|kearsley}} that can be constructed in this way from a [[semi-ellipse]]. To meet the curvature condition, the semi-ellipse should be bounded by the [[Semi-major and semi-minor axes|semi-major axis]] of its ellipse, and the ellipse should have [[Eccentricity (mathematics)|eccentricity]] at most <math>\tfrac{1}{2}\sqrt{3}</math>. Equivalently, the semi-major axis should be at most twice the semi-minor axis.{{r|bs}}

Given any two bodies of constant width, their [[Minkowski sum]] forms another body of constant width.{{r|mmo}}