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 p is defined by , then as soon as the homogeneous polynomial P is not divisible by z. For example, the projective curve of equation x2 + y2 − z2 is the projective completion of the unit circle of equation x2 + y2 − 1 = 0. This implies that an affine curve and its projective completion are the same curves, 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 p(x, y) 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 p(x, y) = 0 at a point (a, b) is Remarkable points of a plane curve In this section, we consider a plane algebraic curve defined by a bivariate polynomial p(x, y) and its projective completion, defined by the homogenization of p. Intersection with a line Knowing the points of intersection of a curve with a given line is frequently useful. The intersection with the axes of coordinates and the asymptotes are useful to draw the curve. Intersecting with a line parallel to the axes allows one to find at least a point in each branch of the curve. If an efficient root-finding algorithm is available, this allows to draw the curve by plotting the intersection point with all the lines parallel to the y-axis and passing through each pixel on the x-axis. If the polynomial defining the curve has a degree d, any line cuts the curve in at most d points. Bézout's theorem asserts that this number is exactly d, if the points are searched in the projective plane over an algebraically closed field (for example the complex numbers), and counted with their multiplicity. The method of computation that follows proves again this theorem, in this simple