Barycentric Coordinates -- from Wolfram MathWorld

Barycentric Coordinates

Barycentric coordinates are triples of numbers corresponding to masses placed at the vertices of a reference triangle . These masses then determine a point , which is the geometric centroid of the three masses and is identified with coordinates . The vertices of the triangle are given by , , and . Barycentric coordinates were discovered by Möbius in 1827 (Coxeter 1969, p. 217; Fauvel et al. 1993).

To find the barycentric coordinates for an arbitrary point , find and from the point at the intersection of the line with the side , and then determine as the mass at that will balance a mass at , thus making the centroid (left figure). Furthermore, the areas of the triangles , , and are proportional to the barycentric coordinates , , and of (right figure; Coxeter 1969, p. 217).

Barycentric coordinates are homogeneous, so

(1)

for .

Barycentric coordinates normalized so that they become the actual areas of the subtriangles are called homogeneous barycentric coordinates. Barycentric coordinates normalized so that

(2)

so that the coordinates give the areas of the subtriangles normalized by the area of the original triangle are called areal coordinates (Coxeter 1969, p. 218). Barycentric and areal coordinates can provide particularly elegant proofs of geometric theorems such as Routh's theorem, Ceva's theorem, and Menelaus' theorem (Coxeter 1969, pp. 219-221).

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