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In mathematics, barycentric coordinates are coordinates defined by the vertices of a simplex. Barycentric coordinates are a form of homogeneous coordinates.
Let x1, ..., xn be the vertices of a simplex in a vector space A. If, for some point p in A,
- <math> (a_1 + \cdots + a_n)\, p = a_1 \, x_1 + \cdots + a_n \, x_n <math>
then we say that the coefficients (a1, ..., an) are barycentric coordinates of p with respect to x1, ..., xn. The vertices themselves have the coordinates (1, 0, 0, ..., 0), (0, 1, 0, ..., 0), ..., (0, 0, 0, ..., 1). Barycentric coordinates are not unique: for any b not equal to zero, (b a1, ..., b an) are also barycentric coordinates of p.
When the coordinates are positive and sum to 1, the point p lies in the convex hull of x1, ..., xn, that is, in the simplex which has those points as its vertices.
If we imagine masses equal to a1, ..., an attached to the vertices of the simplex, the center of gravity (the barycenter) is then p. This is the origin of the term "barycentric", introduced (1827) by August Ferdinand Möbius.
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