Curve of constant width: Difference between revisions

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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 formed from 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. For any long enough line segment, there is a starting position tangent to one of the cusps of the curve for which it will return in this way, obtained by a calculation involving an alternating sum of the lengths of arcs of the starting curve between its cusps.{{r|lowry}}

Another construction chooses half of the curve of constant width, meeting certain conditions, and then completes it to a full curve. The construction begins with a convex curved arc, connecting a pair of closest points on two parallel lines whose separation is the intended width <math>w</math> of the curve. The arc must have the property (required of a curve of constant width) that each of its supporting lines is tangent to a circle of radius <math>w</math> containing the entire arc; intuitively, this prevents its [[curvature]] from being smaller than that of a circle of radius <math>w</math> at any point. As long as it meets this condition, it can be used in the construction. The next step is to intersect an infinite family of circular disks of radius <math>w</math>, both the ones tangent to the supporting lines and additional disks centered at each point of the arc. This intersection forms a body of constant width, with the given arc as part of its boundary. This construction is universal: all curves of constant width may be constructed in this way.{{r|rt}} In a special case of this construction found by 19th-century French mathematician [[Victor Puiseux]],{{r|kearsley}} it can be applied to the arc formed by half of an [[ellipse]] between the ends of its two [[Semi-major and semi-minor axes|semi-major axes]], as long as its [[Eccentricity (mathematics)|eccentricity]] is at most <math>\tfrac{1}{2}\sqrt{3}</math>, low enough to meet the curvature condition. (Equivalently, the semi-major axis should be at most twice the semi-minor axis.){{r|bs}} This construction is universal: all curves of constant width may be constructed in this way.{{r|rt}}

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