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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==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}}