Editing Reuleaux triangle (section)

== Related figures ==
[[File:Triquetra-Vesica.svg|thumb|upright|[[Triquetra]] interlaced to form a [[trefoil knot]]]]
In the classical presentation of a three-set [[Venn diagram]] as three overlapping circles, the central region (representing elements belonging to all three sets) takes the shape of a Reuleaux triangle.<ref name="icons" /> The same three circles form one of the standard drawings of the [[Borromean rings]], three mutually linked rings that cannot, however, be realized as geometric circles.<ref>{{citation
 | last1 = Lindström | first1 = Bernt
 | last2 = Zetterström | first2 = Hans-Olov
 | doi = 10.2307/2323803
 | issue = 4
 | journal = [[American Mathematical Monthly]]
 | jstor = 2323803
 | pages = 340–341
 | title = Borromean circles are impossible
 | volume = 98
 | year = 1991}}.</ref> Parts of these same circles are used to form the [[triquetra]], a figure of three overlapping [[semicircle]]s (each two of which form a [[vesica piscis]] symbol) that again has a Reuleaux triangle at its center;<ref>{{mathworld|id=Triquetra|title=Triquetra|mode=cs2}}</ref> just as the three circles of the Venn diagram may be interlaced to form the Borromean rings, the three circular arcs of the triquetra may be interlaced to form a [[trefoil knot]].<ref>{{citation
 | last1 = Hoy | first1 = Jessica
 | last2 = Millett | first2 = Kenneth C.
 | journal = Journal of Mathematics and the Arts
 | title = A mathematical analysis of knotting and linking in Leonardo da Vinci's cartelle of the Accademia Vinciana
 | url = http://www.math.ucsb.edu/~millett/Papers/Millett2014Leonardov5.pdf
 | year = 2014}}.</ref>

Relatives of the Reuleaux triangle arise in the problem of finding the minimum perimeter shape that encloses a fixed amount of area and includes three specified points in the plane. For a wide range of choices of the area parameter, the optimal solution to this problem will be a curved triangle whose three sides are circular arcs with equal radii. In particular, when the three points are equidistant from each other and the area is that of the Reuleaux triangle, the Reuleaux triangle is the optimal enclosure.<ref>{{citation|title=What Is Mathematics? An Elementary Approach to Ideas and Methods|first1=Richard|last1=Courant|author1-link=Richard Courant|first2=Herbert|last2=Robbins|author2-link=Herbert Robbins|edition=2nd|publisher=Oxford University Press|year=1996|isbn=978-0-19-975487-8|pages=378–379|url=https://books.google.com/books?id=UfdossHPlkgC&pg=PA378}}.</ref>

[[Circular triangle]]s are triangles with circular-arc edges, including the Reuleaux triangle as well as other shapes.
The [[deltoid curve]] is another type of curvilinear triangle, but one in which the curves replacing each side of an equilateral triangle are concave rather than convex. It is not composed of circular arcs, but may be formed by rolling one circle within another of three times the radius.<ref>{{citation | first=E. H.|last=Lockwood| title=A Book of Curves | publisher=Cambridge University Press | year=1961| chapter=Chapter 8: The Deltoid }}</ref> Other planar shapes with three curved sides include the [[arbelos]], which is formed from three [[semicircle]]s with collinear endpoints,<ref>{{citation
 | last = Mackay | first = J. S.
 | date = February 1884
 | doi = 10.1017/s0013091500037196
 | journal = Proceedings of the Edinburgh Mathematical Society
 | page = 2
 | title = The shoemaker's knife
 | volume = 3| doi-access = free
 }}.</ref> and the [[Bézier triangle]].<ref>{{citation
 | last = Bruijns | first = J.
 | contribution = Quadratic Bezier triangles as drawing primitives
 | doi = 10.1145/285305.285307
 | isbn = 978-1-58113-097-3
 | location = New York, NY, USA
 | pages = 15–24
 | publisher = ACM
 | title = Proceedings of the ACM SIGGRAPH/EUROGRAPHICS Workshop on Graphics Hardware (HWWS '98)
 | year = 1998| s2cid = 28967106
 }}.</ref>

The Reuleaux triangle may also be interpreted as the [[stereographic projection]] of one [[spherical triangle|triangular]] face of a [[spherical tetrahedron]], the [[Schwarz triangle]] of parameters <math>\tfrac32, \tfrac32, \tfrac32</math> with [[dihedral angle|spherical angles]] of measure <math>120^\circ</math> and sides of [[central angle|spherical length]] {{nobr|<math>{\arccos}\bigl({-\tfrac13}\bigr).</math><ref name="foam" /><ref>{{citation|title=Spherical Models|first=Magnus J.|last=Wenninger|author-link=Magnus Wenninger|year=2014|publisher=Dover|isbn=978-0-486-14365-1|page=134|url=https://books.google.com/books?id=0cfAAwAAQBAJ&pg=PA134}}.</ref>}}