Barycentric coordinates

In geometry, barycentric coordinates form a homogeneous coordinate system based on a reference simplex. The location of a point with respect to this system is given by the masses (by definition positive) which would need to be placed at the reference points in order to have the given point as barycentre. However, for a point outside the simplex some coordinates must be negative.

In an affine space or vector space of dimension n we take n+1 points $s_{0},s_{1},\ldots ,s_{n}$ in general position (no k+1 of them lie in an affine subspace of dimension less than k) as a simplex of reference. The barycentric coordinates of a point x are an (n+1)-tuple $x_{0},x_{1},\ldots ,x_{n}$ such that

$(x_{0}+\cdots +x_{n})x=x_{0}s_{0}+\cdots +x_{n}s_{n}\,$

and at least one of $x_{0},x_{1},\ldots ,x_{n}$ does not vanish. The coordinates are not affected by scaling, and it may be convenient to take $x_{0}+\cdots +x_{n}=1$, in which case they are called areal coordinates.