barycentric

Barycentric is an adjective used in mathematics and related fields to describe things defined with respect to a barycenter or to barycentric coordinates. In geometry, barycentric coordinates describe a point as an affine combination of the vertices of a reference simplex, typically a triangle in the plane. If A, B, C are the triangle's vertices and P is a point with barycentric coordinates α, β, γ, then α + β + γ = 1 and P = αA + βB + γC. The weights α, β, γ are proportional to the areas of the subtriangles PBC, PCA, and PAB, and are often written as area coordinates.

Barycentric coordinates generalize to higher dimensions: for a simplex with vertices v0, ..., vn, a point P inside with coordinates w0, ..., wn satisfies sum wi = 1 and P = sum wi vi. Nonnegative weights place P inside the simplex; negative weights place P outside. These coordinates are used for affine interpolation within simplices, such as texture mapping and finite element methods.

In physics and geometry, the term relates to the barycenter or center of mass of a system of particles. The barycenter is the weighted average of the positions of the particles with weights given by their masses, and barycentric coordinates can be interpreted as normalized masses.

Barycentric coordinates are distinct from trilinear coordinates, which measure distances to the sides rather than areas. The term also appears in other contexts, such as barycentric subdivision in topology, which centers a simplex at its barycenter to create a refinement.