Real Projective Plane - Homogeneous Coordinates

Homogeneous Coordinates

The points in the plane can be represented by homogeneous coordinates. A point has homogeneous coordinates, where the coordinates and are considered to represent the same point, for all nonzero values of t. The points with coordinates are the usual real plane, called the finite part of the projective plane, and points with coordinates, called points at infinity or ideal points, constitute a line called the line at infinity. (The homogeneous coordinates do not represent any point.)

The lines in the plane can also be represented by homogeneous coordinates. A projective line corresponding to the plane ax + by + cz = 0 in R3 has the homogeneous coordinates (a : b : c). Thus, these coordinates have the equivalence relation (a : b : c) = (da : db : dc) for all nonzero values of d. Hence a different equation of the same line dax + dby + dcz = 0 gives the same homogeneous coordinates. A point lies on a line (a : b : c) if ax + by + cz = 0. Therefore, lines with coordinates (a : b : c) where a, b are not both 0 correspond to the lines in the usual real plane, because they contain points that are not at infinity. The line with coordinates (0 : 0 : 1) is the line at infinity, since the only points on it are those with z = 0.