Algebraic Curve - Plane Projective Curves

Plane Projective Curves

It is often desirable to consider curves in the projective space. An algebraic curve in the projective plane or plane projective curve is the set of the points in a projective plane whose projective coordinates are zeros of a homogeneous polynomial in three variables

Every affine algebraic curve of equation may be completed into the projective curve of equation where is the result of the homogenization of p. Conversely, if is the homogeneous equation of a projective curve, then is the equation of the restriction of the projective curve to the affine plane of the points whose third projective coordinate is not zero. These two operations are reciprocal one to the other, as and as soon as the homogeneous polynomial P is not divisible by z.

For example, the projective curve of equation is the projective completion of the unit circle of equation

This allows to consider that an affine curve and its projective completion are the same curve, or, more precisely that the affine curve is a part of the projective curve that is large enough to well define the "complete" curve. This point of view is commonly expressed by calling "points at infinity" of the affine curve the points (in finite number) of the projective completion that do not belong to the affine part.

Projective curves are frequently studied for themselves. They are also useful for the study of affine curves. For example, if is the polynomial defining an affine curve, beside the partial derivatives and, it is useful to consider the derivative at infinity For example, the equation of the tangent of the affine curve of equation at a point (a, b) is