Editing Reuleaux triangle (section)

=={{anchor|Generalisations}} Generalizations ==
Triangular curves of constant width with smooth rather than sharp corners may be obtained as the locus of points at a fixed distance from the Reuleaux triangle.<ref>{{citation
 | last1 = Banchoff | first1 = Thomas
 | last2 = Giblin | first2 = Peter
 | doi = 10.2307/2974900
 | issue = 5
 | journal = [[American Mathematical Monthly]]
 | mr = 1272938
 | pages = 403–416
 | title = On the geometry of piecewise circular curves
 | volume = 101
 | year = 1994| jstor = 2974900
 }}.</ref> Other generalizations of the Reuleaux triangle include surfaces in three dimensions, curves of constant width with more than three sides, and the Yanmouti sets which provide extreme examples of an inequality between width, diameter, and inradius.

=== Three-dimensional version === <!-- This section is linked from [[Sphere]] -->
[[File:Reuleaux-tetrahedron-intersection.png|thumb|Four balls intersect to form a Reuleaux tetrahedron.]]
The intersection of four [[ball (mathematics)|balls]] of radius ''s'' centered at the vertices of a regular [[tetrahedron]] with side length ''s'' is called the [[Reuleaux tetrahedron]], but its surface is not a [[surface of constant width]].<ref name=weber>{{citation
  | last = Weber | first = Christof
  | year = 2009
  | url = http://www.swisseduc.ch/mathematik/geometrie/gleichdick/docs/meissner_en.pdf
  | title = What does this solid have to do with a ball?}} Weber also has [http://www.swisseduc.ch/mathematik/geometrie/gleichdick/index-en.html films of both types of Meissner body rotating] as well as [http://www.swisseduc.ch/mathematik/geometrie/gleichdick/meissner-en.html interactive  images].</ref>  It can, however, be made into a surface of constant width, called [[Reuleaux tetrahedron#Meissner bodies|Meissner's tetrahedron]], by replacing three of its edge arcs by curved surfaces,  the surfaces of rotation of a circular arc. Alternatively, the [[surface of revolution]] of a Reuleaux triangle through one of its symmetry axes forms a surface of constant width, with minimum volume among all known surfaces of revolution of given constant width.<ref name="ccg">{{citation
  | last1 = Campi | first1 = Stefano
  | last2 = Colesanti | first2 = Andrea
  | last3 = Gronchi | first3 = Paolo
  | contribution = Minimum problems for volumes of convex bodies
  | title = Partial Differential Equations and Applications: Collected Papers in Honor of Carlo Pucci
  | publisher = Lecture Notes in Pure and Applied Mathematics, no. 177, Marcel Dekker
  | year = 1996
  | pages = 43–55}}.</ref>

=== Reuleaux polygons ===
{{main|Reuleaux polygon}}
{{multiple image|total_width=360|image1=Reuleaux polygons.svg|caption1=Reuleaux polygons|image2=Gambia 1 dalasi.JPG|caption2=[[Gambian dalasi]] Reuleaux [[heptagon]] coin}}
The Reuleaux triangle can be generalized to regular or irregular polygons with an odd number of sides, yielding a [[Reuleaux polygon]], a curve of constant width formed from circular arcs of constant radius. The constant width of these shapes allows their use as coins that can be used in coin-operated machines.<ref name="gardner" /> Although coins of this type in general circulation usually have more than three sides, a Reuleaux triangle has been used for a commemorative coin from [[Bermuda]].<ref name=conti>{{citation|last1=Conti|first1=Giuseppe|last2=Paoletti|first2=Raffaella|editor1-last=Magnaghi-Delfino|editor1-first=Paola|editor2-last=Mele|editor2-first=Giampiero|editor3-last=Norando|editor3-first=Tullia|contribution=Reuleaux triangle in architecture and applications|date=October 2019|doi=10.1007/978-3-030-29796-1_7|pages=79–89|publisher=Springer|series=Lecture Notes in Networks and Systems|title=Faces of Geometry: From Agnesi to Mirzakhani|volume=88 |isbn=978-3-030-29795-4 |s2cid=209976466}}</ref>

Similar methods can be used to enclose an arbitrary [[simple polygon]] within a curve of constant width, whose width equals the diameter of the given polygon. The resulting shape consists of circular arcs (at most as many as sides of the polygon), can be constructed algorithmically in [[linear time]], and can be drawn with compass and straightedge.<ref>{{citation
 | last1 = Chandru | first1 = V.
 | last2 = Venkataraman | first2 = R.
 | contribution = Circular hulls and orbiforms of simple polygons
 | contribution-url = http://dl.acm.org/citation.cfm?id=127787.127863
 | isbn = 978-0-89791-376-8
 | location = Philadelphia, PA, USA
 | pages = 433–440
 | publisher = Society for Industrial and Applied Mathematics
 | title = Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '91)
 | year = 1991}}.</ref> Although the Reuleaux polygons all have an odd number of circular-arc sides, it is possible to construct constant-width shapes with an even number of circular-arc sides of varying radii.<ref>{{citation
 | last = Peterson | first = Bruce B.
 | journal = Illinois Journal of Mathematics
 | mr = 0320885
 | pages = 411–420
 | title = Intersection properties of curves of constant width
 | url = http://projecteuclid.org/euclid.ijm/1256051608
 | volume = 17
 | issue = 3
 | year = 1973| doi = 10.1215/ijm/1256051608
 | doi-access = free
 }}.</ref>

=== Yanmouti sets ===
The Yanmouti sets are defined as the [[convex hull]]s of an equilateral triangle together with three circular arcs, centered at the triangle vertices and spanning the same angle as the triangle, with equal radii that are at most equal to the side length of the triangle. Thus, when the radius is small enough, these sets degenerate to the equilateral triangle itself, but when the radius is as large as possible they equal the corresponding Reuleaux triangle. Every shape with width ''w'', diameter ''d'', and inradius ''r'' (the radius of the largest possible circle contained in the shape) obeys the inequality
:<math>w - r \le \frac{d}{\sqrt 3},</math>
and this inequality becomes an equality for the Yanmouti sets, showing that it cannot be improved.<ref>{{citation
 | last = Hernández Cifre | first = M. A.
 | doi = 10.2307/2695582
 | issue = 10
 | journal = [[American Mathematical Monthly]]
 | mr = 1806918
 | pages = 893–900
 | title = Is there a planar convex set with given width, diameter, and inradius?
 | volume = 107
 | year = 2000| jstor = 2695582
 }}.</ref>