Barycentric Coordinates in Triangles: Definition, Applications, and Examples

Barycentric coordinates in a triangle ABC represent a point P as a combination of the three vertices A, B, C, where each vertex is given a weight or coefficient that sums up to 1.

Let P be a point in triangle ABC, then the barycentric coordinates of P with respect to triangle ABC are denoted by (u,v,w) and are defined as follows:

u = [ABC]/[PBC], v = [ACB]/[PAC], w = [BAC]/[PAB]

where [ABC] is the area of triangle ABC and [PBC], [ACB], and [BAC] are the areas of triangles PBC, PAC, and PAB respectively.

Geometrically, the barycentric coordinates of P represent the ratios of the distances from P to the vertices of the triangle ABC. For example, u represents the ratio of the distance from P to B over the distance from A to B.

Barycentric coordinates have many applications in geometry, including triangle geometry, projective geometry, and computational geometry. They are also used in computer graphics and computer-aided design.