Curve of constant width: Difference between revisions

Revision as of 05:33, 12 January 2021

Latest revision as of 18:20, 13 August 2024

(26 intermediate revisions by 9 users not shown)

Line 1:

{{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.

Line 26 ⟶ 27:

&+ 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==

Line 32 ⟶ 33:

[[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 red circle shows a tangent circle to a supporting line, at a [[Vertex (curve)|point of minimum curvature]] of the semi-ellipse. The eccentricity of the semi-ellipse in the figure is the maximum possible for this construction.]]

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 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}}

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==

Line 50 ⟶ 51:

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 together touch the curve in at least three pairs of opposite points, but these points are not necessarily 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==

Line 56 ⟶ 57:

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}}

Line 77 ⟶ 78:

| 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

| year = 1860}} See in particular pp. 283–285.</ref>

<ref name=bb>{{cite arxivarXiv

| last1 = Bardet | first1 = Magali

| last2 = Bayen | first2 = Térence

| arxiveprint = 1312.4358

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

| year = 2013}}</ref>| class = math.AG

}}</ref>

<ref name=bs>{{cite book

Line 119 ⟶ 121:

| url = https://projecteuclid.org/euclid.pjm/1102993951

| volume = 19

| year = 1966}}<| doi = 10.2140/ref>pjm.1966.19.13

| doi-access = free

}}</ref>

<ref name=chamberland>{{cite book

Line 147 ⟶ 151:

| publisher = Oxford University Press

| title = Mathematical Models

| title-link = Mathematical Models (Cundy and Rollett)

| year = 1961}}</ref>

Line 161 ⟶ 166:

| volume = 185

| year = 2018| arxiv = 1608.01651

| s2cid = 119710622

}}</ref>

Line 171 ⟶ 177:

| title = Sets of constant width in finite dimensional Banach spaces

| volume = 3

| year = 1965}}</ref>| issue = 3

| s2cid = 121731141

}}</ref>

<ref name=euler>{{cite journal

Line 187 ⟶ 195:

| last = Fujiwara | first = M. | authorlink = Matsusaburo Fujiwara

| journal = [[Tohoku Mathematical Journal]]

| pages = 180-184180–184

| series = 1st series

| title = On space curves of constant breadth

Line 232 ⟶ 240:

| title = Über konvexe Punktmengen konstanter Breite

| volume = 29

| year = 1929| s2cid = 122800988 }}</ref>

<ref name=kearsley>{{cite journal

Line 243 ⟶ 251:

| pages = 176–179

| title = Curves of constant diameter

| volume = 36}}</ref>| s2cid = 125468725

}}</ref>

<ref name=kelly>{{cite journal

| last = Kelly | first = Paul J. | author-link = Paul Kelly (mathematician)

| doi = 10.2307/2309594

| journal = [[American Mathematical Monthly]]

| jstor = 2309594

| mr = 92168

| pages = 333–336

| title = Curves with a kind of constant width

| volume = 64

| year = 1957| issue = 5 }}</ref>

<ref name=lay>{{cite book

Line 262 ⟶ 282:

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

| volume = 75

| year = 2005}}</ref>| s2cid = 119927817

}}</ref>

<ref name=lowry>{{cite journal

Line 274 ⟶ 295:

| page = 43

| title = 2109. Curves of constant diameter

| volume = 34}}</ref>| s2cid = 187767688

}}</ref>

<ref name=martinez>{{cite journal

Line 300 ⟶ 322:

| last1 = Martini | first1 = Horst

| last2 = Montejano | first2 = Luis

| last3 = Oliveros | first3 = Déborah | author3-link = Déborah Oliveros

| doi = 10.1007/978-3-030-03868-7

| isbn = 978-3-030-03866-3

Line 306 ⟶ 328:

| publisher = Birkhäuser

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

| year = 2019| s2cid = 127264210
}} 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>

<ref name=moore>{{cite book

Line 327 ⟶ 350:

| journal = Missouri Journal of Mathematical Sciences

| mr = 1455287

|doi=10.35834/1997/0901023

|doi-access=free

| pages = 23–27

| title = A polynomial curve of constant width

Line 373 ⟶ 398:

| title = Globale Sätze über Raumkurven konstanter Breite

| volume = 53

| year = 1972}}</ref>| issue = 1–6

}}</ref>

<ref name=wegner77>{{cite journal

Line 391 ⟶ 417:

==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}}