Subject 6.04: What are barycentric coordinates?
Let p1, p2, p3 be the three vertices (corners) of a closed triangle T (in 2D or 3D or any dimension). Let t1, t2, t3 be real numbers that sum to 1, with each between 0 and 1: t1 + t2 + t3 = 1, 0 <= ti <= 1. Then the point p = t1*p1 + t2*p2 + t3*p3 lies in the plane of T and is inside T. The numbers (t1,t2,t3) are called the barycentric coordinates of p with respect to T. They uniquely identify p. They can be viewed as masses placed at the vertices whose center of gravity is p. For example, let p1=(0,0), p2=(1,0), p3=(3,2). The barycentric coordinates (1/2,0,1/2) specify the point p = (0,0)/2 + 0*(1,0) + (3,2)/2 = (3/2,1), the midpoint of the p1-p3 edge. If p is joined to the three vertices, T is partitioned into three triangles. Call them T1, T2, T3, with Ti not incident to pi. The areas of these triangles Ti are proportional to the barycentric coordinates ti of p. Reference: [Coxeter, Intro. to Geometry, p.217].
