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==

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[[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 from [[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}}

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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==

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==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}}

One way to generalize these concepts to three dimensions is through the [[surface of constant width|surfaces of constant width]]. The three-dimensional analog of a Reuleaux triangle, the [[Reuleaux tetrahedron]], does not have constant width, but minor changes to it produce the [[Meissner bodies]], which do.{{r|gardner|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}}

One way to generalize these concepts to three dimensions is through the [[surface of constant width|surfaces of constant width]]. The three-dimensional analog of a Reuleaux triangle, the [[Reuleaux tetrahedron]], does not have constant width, but minor changes to it produce the [[Meissner bodies]], which do.{{r|gardner|mmo}} The curves 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 generalization 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

| 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

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| 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

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| publisher = Oxford University Press

| title = Mathematical Models

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

| year = 1961}}</ref>

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| volume = 185

| year = 2018| arxiv = 1608.01651

| s2cid = 119710622

}}</ref>

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| 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

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| 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

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| title = Über konvexe Punktmengen konstanter Breite

| volume = 29

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

<ref name=kearsley>{{cite journal

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| 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

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

| volume = 75

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

}}</ref>

<ref name=lowry>{{cite journal

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| page = 43

| title = 2109. Curves of constant diameter

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

}}</ref>

<ref name=martinez>{{cite journal

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| 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

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| 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

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| 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

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| title = Globale Sätze über Raumkurven konstanter Breite

| volume = 53

| year = 1972}}</ref>| issue = 1–6

}}</ref>

<ref name=wegner77>{{cite journal

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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}}