 Barycentric coordinate system

A barycentric coordinate system assigns to a point in the affine span of the vertices of a nondegenerate simplex its unique vertex weights ${\lambda }_0,\dots ,{\lambda }_d$, constrained to sum to one. Uniqueness follows because $v_1-v_0,\dots ,v_d-v_0$ form a basis of the translation space: expand $x-v_0$ in that basis and set ${\lambda }_0=1-{\sum }_{i=1}^d{\lambda }_i$. A point lies in the closed simplex exactly when every weight is nonnegative. An affine function satisfies $f(x)={\sum }_i{\lambda }_{i}f(v_i)$, which makes these coordinates useful for interpolation and the integral of an affine function over a simplex.

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.