Curve of constant width: Difference between revisions

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{{short description|ConvexShape planar shape whosewith width is the same regardlessindependent of the orientation of the curve}}

{{good article}}

[[File:Reuleaux supporting lines.svg|thumb|Measuring the width of a [[Reuleaux triangle]] as the distance between parallel [[supporting line]]s. Because this distance does not depend on the direction of the lines, the Reuleaux triangle is a curve of constant width.]]

In [[geometry]], a '''curve of constant width''' is a [[simple closed curve]] in the [[plane (geometry)|plane]] whose width (the distance between parallel [[supporting line]]s) is the same in all directions. The shape bounded by a curve of constant width is a '''body of constant width''' or an '''orbiform''', the name given to these shapes by [[Leonhard Euler]].{{r|euler}} Standard examples are the [[circle]] and the [[Reuleaux triangle]]. These curves can also be constructed using circular arcs centered at crossings of an [[arrangement of lines]], as the [[involute]]s of certain curves, or by intersecting circles centered on a partial curve.

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&+ x(x^2 - 3y^2)\left(16(x^2 + y^2)^2 - 5544(x^2 + y^2) + 266382\right) - 720^3.

\end{align}</math>

Its [[Degree of a polynomial|degree]], eight, is the minimum possible degree for a polynomial that defines a non-circular curve of constant width.{{r|bb}}

==Constructions==

[[File:Reuleaux polygon construction.svg|thumb|upright=0.8|An irregular [[Reuleaux polygon]]]]

[[File:Crossed-lines constant-width.svg|thumb|Applying the crossed-lines method to an [[arrangement of lines|arrangement of four lines]]. The boundaries of the blue body of constant width are circular arcs from four nested pairs of circles (inner circles dark red and outer circles light red).]]

[[File:Constant-width semi-ellipse.svg|thumb|upright=1.3|Body of constant width (yellow) formed by intersecting disks (blue) centered on a [[semi-ellipse]] (black). The eccentricityred ofcircle theshows semi-ellipsea intangent thecircle figureto is the maximum possible while maintaining the property that everya supporting line, isat tangenta to[[Vertex a circle(curve)|point of radiusminimum equalcurvature]] to the width, containingof the semi-ellipse;. thisThe tangenteccentricity circleof isthe shownsemi-ellipse in red for the [[Vertexfigure (curve)|pointis ofthe minimummaximum curvature]]possible offor thethis semi-ellipseconstruction.]]

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 ofIn a moreclosely generalrelated construction, called by [[Martin Gardner]] the "crossed-lines method", in which anyan [[arrangement of lines]] in the plane (no two parallel but otherwise arbitrary), is sorted into cyclic order by theirthe slopes, of the lines. The lines are then connected by a smooth curve formed from a sequence of circular arcs; betweeneach pairsarc ofconnects two consecutive lines in the sorted order, and is centered at thetheir 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, with constant width equal to the length of the line segment.{{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 iany 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 conditionsrequirements, and thenforms completesfrom it to a fullbody of constant width having the given curve as part of its boundary. The construction begins with a convex curved arc, connectingwhose aendpoints pairare ofthe closestintended pointswidth on<math>w</math> apart. The two parallelendpoints linesmust whosetouch separationparallel issupporting thelines intendedat widthdistance <math>w</math> offrom theeach curveother. TheAdditionally, arceach mustsupporting haveline thethat propertytouches (requiredanother of a curvepoint of constantthe width)arc thatmust eachbe oftangent itsat supportingthat lines is tangentpoint to a circle of radius <math>w</math> containing the entire arc; intuitively, this requirement prevents itsthe [[curvature]] of the arc from being smallerless than that of athe circle of radius <math>w</math> at any point. AsThe longcompleted asbody itof meetsconstant thiswidth condition,is itthen canthe beintersection used inof the construction.interiors The next step is to intersectof an infinite family of circular diskscircles, of radiustwo <math>w</math>, bothtypes: the ones tangent to the supporting lines, and additionalmore disks centered at each pointcircles of the arc.same Thisradius intersectioncentered formsat aeach bodypoint 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}} [[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, whichand 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}} A generalization of Minkowski sums to the sums of support functions of hedgehogs produces a curve of constant width from the sum of a projective hedgehog and a circle, whenever the result is a convex curve. All curves of constant width can be decomposed into a sum of hedgehogs in this way.{{r|martinez}}

==Properties==

[[File:Reuleaux triangle Animation.gif|thumb|The Reuleaux triangle rolling within a square while at all times touching all four sides]]

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 of constant width can be rotated between two parallel lines separated by its width, while at all times during the rotation touching those lines.

This sequence of rotations of the curve can be obtained by keeping the curve fixed in place and rotating two supporting lines around it, and then applying rotations of the whole plane that instead keep the lines in place and cause the curve to rotate between them. In the same way, a curve of constant width can be rotated between two pairs of parallel lines with the same separation. In particular, by choosing the lines through opposite sides of a [[Square (geometry)|square]], any curve of constant width can be rotated within a square.{{r|gardner|bs|rt}} Although it is not always possible to rotate such a curve within a regular [[hexagon]], every curve of constant width can be drawn within a regular hexagon in such a way that it touches all six sides.{{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}}

A [[vertex (curve)|vertex of a smooth curve]] is a point where its curvature is a local maximum or minimum; for a circular arc, all points are vertices, but non-circular curves may have a finite discrete set of vertices. For a curve that is not smooth, the points where it is not smooth can also be considered as vertices, of infinite curvature. For a curve of constant width, each vertex of locally minimum curvature is paired with a vertex of locally maximum curvature, opposite it on a diameter of the curve, and there must be at least six vertices. This stands in contrast to the [[four-vertex theorem]], according to which every simple closed smooth curve in the plane has at least four vertices. Some curves, such as ellipses, have exactly four vertices, but this is not possible for a curve of constant width.{{r|martinez|ctb}} Because local minima of curvature are opposite local maxima of curvature, the only curves of constant width with [[central symmetry]] are the circles, for which the curvature is the same at all points.{{r|mmo}} For every curve of constant width, the [[Smallest-circle problem|minimum enclosing circle]] of the curve and the largest circle that it contains are concentric, and the average of their diameters is the width of the curve. These two circles againtogether touch the curve in at least three pairs of opposite points, but these touching points mightare not benecessarily vertices.{{r|mmo}}

A convex body has constant width if and only if the Minkowski sum of the body and its central180° reflectionrotation is a circular disk; if so, the width of the body is the radius of the disk.{{r|mmo|chakerian}}

==Applications==

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Because of the ability of curves of constant width to roll between parallel lines, any [[cylinder]] with a curve of constant width as its cross-section can act as a [[Bearing (mechanical)#History|"roller"]], supporting a level plane and keeping it flat as it rolls along any level surface. However, the center of the roller moves up and down as it rolls, so this construction would not work for wheels in this shape attached to fixed axles.{{r|gardner|bs|rt}}

Several countries haveSome [[Coinagecoinage shapes|coins shaped]] asare non-circular curvesbodies of constant width;. examplesFor includeinstance the British [[British coin Twenty Pence|20p]] and [[British coin Fifty Pence|50p]] coins. Theirare heptagonalReuleaux shapeheptagons, with curved sides means thatand the Canadian [[currency detectorloonie]] inis ana Reuleaux 11-gon.{{r|chamberland}} These shapes allow automated coin machinemachines willto alwaysrecognize measurethese thecoins samefrom widththeir widths, noregardless matterof whichthe angleorientation itof takesthe itscoin measurementin fromthe machine.{{r|gardner|bs}} TheOn samethe other hand, testing the width is trueinadequate ofto determine the 11-sided[[Roundness (object)|roundness of an object]], because such tests cannot distinguish circles from other curves of constant width.{{r|gardner|bs}} Overlooking this fact may have played a role in the [[loonieSpace Shuttle Challenger disaster]], (Canadianas dollarthe coin)roundness of sections of the rocket in that launch was tested only by measuring widths, and off-round shapes may cause unusually high stresses that could have been one of the factors causing the disaster.{{r|chamberlandmoore}}

Because of the existence of non-circular curves of constant width, checking the [[Roundness (object)|roundness of an object]] requires more complex measurements than its width.{{r|gardner|bs}} Overlooking this fact may have played a role in the [[Space Shuttle Challenger disaster]], as the roundness of sections of the rocket in that launch was tested only by measuring different diameters, and off-round shapes may cause unusually high stresses that could have been one of the factors causing the disaster.{{r|moore}}

==Generalizations==

The curves of constant width can be generalized to certain non-convex curves, the curves that have two tangent lines in each direction, with the same separation between these two lines regardless of their direction. As a limiting case, the [[Hedgehog (geometry)|projective hedgehogs]] (curves with one tangent line in each direction) have also been called "curves of zero width".{{r|kelly}}

The generalization of the definition of bodies of constant width to convex bodies in <math>\mathbb{R}^3</math> and their boundaries leads to the concept of [[surface of constant width]] (in the case of a Reuleaux triangle, this does not lead to a [[Reuleaux tetrahedron]], but to [[Meissner bodies]]).{{r|gardner|mmo}} There is also a concept of [[space curve]]s of constant width, defined by the properties that each plane that crosses the curve perpendicularly intersects it at exactly one other point, where it is also perpendicular, and that all pairs of points intersected by perpendicular planes are the same distance apart.{{r|fujiwara|cieslak|teufel|wegner72}}

TheOne generalizationway ofto thegeneralize definitionthese of bodies of constant widthconcepts to convexthree bodiesdimensions inis <math>\mathbb{R}^3</math> and their boundaries leads tothrough the concept of [[surface of constant width|surfaces of constant width]]. (inThe thethree-dimensional caseanalog of a Reuleaux triangle, this does not lead to athe [[Reuleaux tetrahedron]], does not have constant width, but minor changes to it produce the [[Meissner bodies]]), which do.{{r|gardner|mmo}} ThereThe iscurves of constant width may also be generalized to the [[Body of constant brightness|bodies of constant brightness]], three-dimensional shapes whose two-dimensional projections all have equal area; these shapes obey a conceptgeneralization of Barbier's theorem.{{r|mmo}} A different class of three-dimensional generalizations, the [[space curve]]s of constant width, are defined by the properties that each plane that crosses the curve perpendicularly intersects it at exactly one other point, where it is also perpendicular, and that all pairs of points intersected by perpendicular planes are the same distance apart.{{r|fujiwara|cieslak|teufel|wegner72}}

Curves and bodies of constant width have also been studied in [[non-Euclidean geometry]]{{r|leichtweiss}} and for non-Euclidean [[normed vector space]]s.{{r|eggleston}}

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| language = French

| pages = 273–286

| title = Note sur le problème de l’aiguillel'aiguille et le jeu du joint couvert

| url = http://portail.mathdoc.fr/JMPA/PDF/JMPA_1860_2_5_A18_0.pdf

| volume = 5

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| title = On the degree of the polynomial defining a planar algebraic curves of constant width

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| title = Über konvexe Punktmengen konstanter Breite

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| pages = 176–179

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| title = Curves of constant width in the non-Euclidean geometry

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| page = 43

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| last1 = Martini | first1 = Horst

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| publisher = Birkhäuser

| title = Bodies of Constant Width: An Introduction to Convex Geometry with Applications

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}} For properties of planar curves of constant width, see in particular pp. 69–71. For the Meissner bodies, see section 8.3, pp. 171–178. For bodies of constant brightness, see section 13.3.2, pp. 310–313.</ref>

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| title = Globale Sätze über Raumkurven konstanter Breite

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==External links==

{{commonscat|Curves of constant width}}

* [http://tube.geogebra.org/m/3597 Interactive Applet] {{Webarchive|url=https://web.archive.org/web/20151123043319/http://tube.geogebra.org/m/3597 |date=2015-11-23 }} by Michael Borcherds showing an irregular shape of constant width (that you can change) made using [http://www.geogebra.org/ GeoGebra].

* {{mathworld|id=CurveofConstantWidth|title=Curve of Constant Width}}

* {{cite web|title=Shapes and Solids of Constant Width|url=http://www.numberphile.com/videos/shapes_constant.html|work=Numberphile|publisher=[[Brady Haran]]|author=Mould, Steve|access-date=2013-11-17|archive-url=https://web.archive.org/web/20160319140111/http://www.numberphile.com/videos/shapes_constant.html|archive-date=2016-03-19|url-status=dead}}