Curve of constant width: Difference between revisions

Revision as of 19:39, 25 January 2022

Latest revision as of 18:20, 13 August 2024

(6 intermediate revisions by 5 users not shown)

Line 1:

{{good article}}

{{short description|Shape with width independent of orientation}}

{{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 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 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 120 ⟶ 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 163 ⟶ 166:

| volume = 185

| year = 2018| arxiv = 1608.01651

| s2cid = 119710622

}}</ref>

Line 173 ⟶ 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 189 ⟶ 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 234 ⟶ 240:

| title = Über konvexe Punktmengen konstanter Breite

| volume = 29

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

<ref name=kearsley>{{cite journal

Line 245 ⟶ 251:

| pages = 176–179

| title = Curves of constant diameter

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

}}</ref>

<ref name=kelly>{{cite journal

Line 256 ⟶ 263:

| title = Curves with a kind of constant width

| volume = 64

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

<ref name=lay>{{cite book

Line 275 ⟶ 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 287 ⟶ 295:

| page = 43

| title = 2109. Curves of constant diameter

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

}}</ref>

<ref name=martinez>{{cite journal

Line 313 ⟶ 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 319 ⟶ 328:

| publisher = Birkhäuser

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

| year = 2019| s2cid = 127264210

| year = 2019}} 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 388 ⟶ 398:

| title = Globale Sätze über Raumkurven konstanter Breite

| volume = 53

| year = 1972}}</ref>| issue = 1–6

}}</ref>

<ref name=wegner77>{{cite journal

Line 406 ⟶ 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}}