Editing Barycentric coordinate system (section)

{{short description|Coordinate system that is defined by points instead of vectors}}
{{Distinguish|Barycentric coordinates (astronomy)}}

[[File:TriangleBarycentricCoordinates.svg|thumb|upright=1.3|Barycentric coordinates <math>(\lambda_1, \lambda_2, \lambda_3)</math> on an equilateral triangle and on a right triangle.]]
[[File:Barycentric subdivision of a 3-simplex.svg|thumb|A 3-simplex, with barycentric subdivisions of 1-faces (edges) 2-faces (triangles) and 3-faces (body).]]

In [[geometry]], a '''barycentric coordinate system''' is a [[coordinate system]] in which the location of a point is specified by reference to a [[simplex]] (a [[triangle]] for points in a [[plane (mathematics)|plane]], a [[tetrahedron]] for points in [[three-dimensional space]], etc.). The '''barycentric coordinates''' of a point can be interpreted as [[mass]]es placed at the vertices of the simplex, such that the point is the [[center of mass]] (or ''barycenter'') of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex.

Every point has barycentric coordinates, and their sum is never zero. Two [[tuple]]s of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number. Therefore, barycentric coordinates are either considered to be defined [[up to]] multiplication by a nonzero constant, or normalized for summing to unity.

Barycentric coordinates were introduced by [[August Ferdinand Möbius|August Möbius]] in 1827.<ref>{{cite book |last=Möbius |first=August Ferdinand |author-link=August Ferdinand Möbius |year=1827 |title=Der barycentrische Calcul |publisher=J.A. Barth |place=Leipzig}}<br>
Reprinted in {{cite book |last=Möbius |first=August Ferdinand |display-authors=0 |year=1885 |contribution=Der barycentrische Calcul |title=August Ferdinand Möbius Gesammelte Werke |volume=1 |editor-last=Baltzer |editor-first=Richard |publisher=S. Hirzel |place=Leipzig |pages=1–388 |contribution-url=https://archive.org/details/gesammeltewerkeh01mbuoft/page/n24/ }}</ref><ref>Max Koecher, Aloys Krieg: ''Ebene Geometrie.'' Springer-Verlag, Berlin 2007, {{ISBN|978-3-540-49328-0}}, S.&nbsp;76.</ref><ref name=Hille>Hille, Einar. "Analytic Function Theory, Volume I", Second edition, fifth printing. Chelsea Publishing Company, New York, 1982,  {{ISBN|0-8284-0269-8}}, page 33, footnote 1</ref> They are special [[homogeneous coordinates]]. Barycentric coordinates are strongly related with [[Cartesian coordinates]] and, more generally, to [[affine coordinates]] ({{xref|see {{slink|Affine space|Relationship between barycentric and affine coordinates}}}}).

Barycentric coordinates are particularly useful in [[triangle geometry]] for studying properties that do not depend on the angles of the triangle, such as [[Ceva's theorem]], [[Routh's theorem]], and [[Menelaus's theorem]]. In [[computer-aided design]], they are useful for defining some kinds of [[Bézier surface]]s.<ref>Josef Hoschek, Dieter Lasser: ''Grundlagen der geometrischen Datenverarbeitung.'' Teubner-Verlag, 1989, {{ISBN|3-519-02962-6}}, S.&nbsp;243.</ref><ref>Gerald Farin: ''Curves and Surfaces for Computer Aided Geometric Design.'' Academic Press, 1990, {{ISBN|9780122490514}}, S.&nbsp;20.</ref>