Curve of constant width: Difference between revisions

Revision as of 08:16, 12 January 2021

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Every [[regular polygon]] with an odd number of sides gives rise to a curve of constant width, a [[Reuleaux polygon]], formed from circular arcs centered at its vertices that pass through the two vertices farthest from the center. For instance, this construction generates a Reuleaux triangle from an equilateral triangle. Some irregular polygons also generate Reuleaux polygons.{{r|bs|cr}} The Reuleaux polygons are a special case of a more general construction, called by [[Martin Gardner]] the "crossed-lines method", in which any [[arrangement of lines]] in the plane (no two parallel), sorted into cyclic order by their slopes, are connected by a smooth curve formed from circular arcs between pairs of consecutive lines in the sorted order, centered at the crossing of these two lines. The radius of the first arc must be chosen large enough to cause all successive arcs to end on the correct side of the next crossing point; however, all sufficiently-large radii work. For two lines, this forms a circle; for three lines on the sides of an equilateral triangle, with the minimum possible radius, it forms a Reuleaux triangle, and for the lines of a regular [[star polygon]] it can form a Reuleaux polygon.{{r|gardner|bs}}

[[Leonhard Euler]] constructed curves of constant width as thefrom [[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]]).{{r|euler|robertson}} An intuitive way to describe the involute construction is to roll a line segment around such a curve, keeping it tangent to the curve without sliding along it, until it returns to its starting point of tangency. The line segment must be long enough to reach the cusp points of the curve, so that it can roll past each cusp to the next part of the curve, and its starting position should be carefully chosen so that at the end of the rolling process it is in the same position it started from. When that happens, the curve traced out by the endpoints of the line segment is an involute that encloses the given curve without crossing it, and is a curve of constant width.{{r|lowry}} 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 the involutes of the deltoid that enclose it 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. Additionally, each supporting line that touches another point of the 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}}