Trilinear Coordinates - Conversions

Conversions

A point with trilinears α : β : γ has barycentric coordinates aα : bβ : cγ where a, b, c are the sidelengths of the triangle. Conversely, a point with barycentrics α : β : γ has trilinears α/a : β/b : γ/c.

There are formulas for converting between trilinear coordinates and 2D Cartesian coordinates. Given a reference triangle ABC express the position of the vertex B in terms of an ordered pair of Cartesian coordinates and represent this algebraically as a vector a using vertex C as the origin. Similarly define the position vector of vertex A as b. Then any point P associated with the reference triangle ABC can be defined in a 2D Cartesian system as a vector P = αa + βb. If this point P has trilinear coordinates x : y : z then the conversion formulas are as follows:

conversely

If an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors A, B and C and if the point P has trilinear coordinates x : y : z, then the Cartesian coordinates of P are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates ax, by and cz as the weights.

Hence

where |C−B| = a, |A−C| = b and |B−A| = c.