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Concept
Barycentric coordinate system
Summary
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, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses 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 not zero. Two tuples 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 Möbius in 1827. They are special homogenous coordinates. Barycentric coordinates are strongly related with Cartesian coordinates and, more generally, to affine coordinates (see ).
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 surfaces.
Let be n + 1 points in a Euclidean space, a flat or an affine space of dimension n that are affinely independent; this means that there is no affine subspace of dimension n - 1 that contains all the points, or, equivalently that the points define a simplex. Given any point there are scalars that are not all zero, such that
for any point O. (As usual, the notation represents the translation vector or free vector that maps the point A to the point B.
Official source
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