Barycentric coordinates

Barycentric coordinates express a point as an affine combination of chosen reference points, with coefficients summing to one. They provide a coordinate system tied to a simplex or other affine frame.

Affine combination: A weighted sum of points whose coefficients sum to one. Barycentric coordinates are the coefficients in this general way of combining points.
Barycentric interpolation in computer graphics: A rasterization method that interpolates vertex attributes across a rendered triangle. Screen-space barycentric weights blend a triangle’s vertex colors, depths, and texture coordinates.
Affine space: A geometric space whose points can be combined affinely, but have no distinguished origin. Barycentric coordinates depend on affine combinations rather than a chosen origin.
Triangle: A polygon with three sides and three vertices. Three vertex weights locate any point in the triangle’s plane when the vertices are noncollinear.
Affine independence: A set of points is affinely independent when none is an affine combination of the others. It determines whether reference points give unique barycentric coordinates.
Finite element method: A numerical method that approximates solutions to differential equations using functions defined on small mesh elements. Element shape functions often use barycentric coordinates to interpolate fields over simplices.
Convex combination: A weighted sum with nonnegative coefficients that sum to one. Nonnegative barycentric coordinates place a point inside the convex hull of the reference points.
Tetrahedron: A polyhedron with four triangular faces and four vertices. Four vertex weights extend triangular barycentric coordinates to three-dimensional space.
Simplex: A generalization of a triangle or tetrahedron to any dimension, formed by affinely independent vertices. Its vertices provide the standard reference points for barycentric coordinates.
Barycentric subdivision: A subdivision of a simplicial complex whose new vertices are barycenters of its faces. The same affine point construction supplies the subdivision’s new vertices.

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