Homogeneous Coordinates - Trilinear Coordinates

Trilinear Coordinates

Let l, m, n be three lines in the plane and define a set of coordinates X, Y and Z of a point p as the signed distances from p to these three lines. These are called the trilinear coordinates of p with respect to the triangle. Strictly speaking these are not homogeneous, since the values of X, Y and Z are determined exactly, not just up to proportionality. There is a linear relationship between them however, so these coordinates can be made homogeneous by allowing multiples of (X, Y, Z) to represent the same point. More generally, X, Y and Z can be defined as constants p, r and q times the distances to l, m and n, resulting in a different system of homogeneous coordinates with the same triangle of reference. This is, in fact, the most general type of system of homogeneous coordinates for points in the plane if none of the lines is the line at infinity.