Homogeneous Coordinates - Change of Coordinate Systems

Change of Coordinate Systems

Just as the selection of axes in the Cartesian coordinate is somewhat arbitrary, the selection of a single system of homogeneous coordinates out of all possible systems is somewhat arbitrary. Therefore it is useful to know how the different systems are related to each other.

Let (x, y, z) be the homogeneous coordinates of a point in the projective plane and for a fixed matrix

with det(A) ≠ 0, define a new set of coordinates (X, Y, Z) by the equation

Multiplication of (x, y, z) by a scalar results in the multiplication of (X, Y, Z) by the same scalar, and X, Y and Z cannot be all 0 unless x, y and z are all zero since A is nonsingular. So (X, Y, Z) are a new system of homogeneous coordinates for points in the projective plane. If z is fixed at 1 then

are proportional to the signed distances from the point to the lines

(The signed distance is the distance multiplied a sign 1 or −1 depending on which side of the line the point lies.) Note that for a = b = 0 the value of X is simply a constant, and similarly for Y and Z.

The three lines,

in homogeneous coordinates, or

in the (X, Y, Z) system, form a triangle called the triangle of reference for the system.