HOMOGENEOUS COORDINATES

Euclidean space R^2 or R^3
-- or Projective Euclidean space P^2 or P^3.

(x,y,z,w) instead of (x,y,z).
-- also written (x:y:z:w)

Homogenizing coordinate: usually the first or last.
-- in OpenGL, it is w.
-- When w=0, this represents a direction or point at infinity.

Advantages of homogeneous coordinates.