Barycentric Coordinates

Barycentric coordinates express any point p in the plane as a unique affine combination of three non-collinear triangle vertices v₁, v₂, and v₃, directly encoding whether p lies strictly inside, on the boundary, or outside the triangle.

Experiment

Explore interior interpolation, edge boundaries, exterior extrapolation, and collinear degeneracy in https://riano.app/module/barycentric-coordinates:

1. Keep the default triangle (0,0), (3,0), (0,3) with p = (1,1) to verify equal weights (1/3, 1/3, 1/3) and classification Inside.
2. Move Point x to 0 and Point y to 0 so p coincides with v₁, yielding weights [1, 0, 0] and classification Boundary.
3. Set collinear vertices (-1,-1), (0,0), (1,1) to verify that the singular system reports Degenerate and hides undefined weights as Unavailable.