The barycentric coordinate system is a coordinate system used in a given triangle (or more generally, in a simplex in higher dimensions) to express the position of a point relative to the vertices of that triangle (or simplex). It is particularly useful in computer graphics, geometric modeling, and finite element analysis.
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A barycentric coordinate system assigns to a point in the affine span of the vertices of a nondegenerate simplex its unique vertex weights , constrained to sum to one. Uniqueness follows because form a basis of the translation space: expand in that basis and set . A point lies in the closed simplex exactly when every weight is nonnegative. An affine function satisfies , which makes these coordinates useful for interpolation and the integral of an affine function over a simplex.