affine algebraic curve of equation may be completed into the projective curve of equation , where

is the result of the homogenization of . Conversely, if is the homogeneous equation of a projective curve, then is the equation of an affine curve, which consists of the points of the projective curve whose third projective coordinate is not zero. These two operations are reciprocal one to the other, as and, if is defined by , then as soon as the homogeneous polynomial is not divisible