Barycentric coordinates

In mathematics, barycentric coordinates are coordinates defined by the vertices of a simplex. Barycentric coordinates are a form of homogeneous coordinates.

Let x₁, ..., x_n be the vertices of a simplex in a vector space A. If, for some point p in A,

$(a_1 + \cdots + a_n)\, p = a_1 \, x_1 + \cdots + a_n \, x_n$

then we say that the coefficients (a₁, ..., a_n) are barycentric coordinates of p with respect to x₁, ..., x_n. 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 a_n) 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, ..., x_n, that is, in the simplex which has those points as its vertices.

If we imagine masses equal to a₁, ..., a_n attached to the vertices of the simplex, the center of mass (the barycenter) is then p. This is the origin of the term "barycentric", introduced (1827) by August Ferdinand Möbius.

External links

Barycentric coordinates: A Curious Application (http://www.cut-the-knot.org/triangle/glasses.shtml) (solving the "three glasses" problem)

de:baryzentrische Koordinaten

Categories: Linear algebra | Geometry

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