Homogeneous Coordinates - Application To Bézout's Theorem

Application To Bézout's Theorem

Bézout's theorem predicts that the number of points of intersection of two curves is equal to the product of their degrees (assuming an algebraically complete field and with certain conventions followed for counting intersection multiplicities). Bézout's theorem predicts there is one point of intersection of two lines and in general this is true, but when the lines are parallel the point of intersection is infinite. Homogeneous coordinates can be used to locate the point of intersection in this case. Similarly, Bézout's theorem predicts that a line will intersect a conic at two points, but in some cases one or both of the points is infinite and homogeneous coordinates must be used to locate them. For example, y = x2 and x = 0 have only one point of intersection in the finite plane. To find the other point of intersection, convert the equations into homogeneous form, yz = x2 and x = 0. This produces x = yz = 0 and, assuming not all of x, y and z are 0, the solutions are x = y = 0, z ≠ 0 and x = z = 0, y ≠ 0. This first solution is the point (0, 0) in Cartesian coordinates, the finite point of intersection. The second solutions gives the homogeneous coordinates (0, 1, 0) which corresponds to the direction of the y-axis. For the equations xy = 1 and x = 0 there are no finite points of intersection. Converting the equations into homogeneous form gives xy = z2 and x = 0. Solving produces the equation z2 = 0 which has a double root at z = 0. From the original equation, x = 0, so y ≠ 0 since at least one coordinate must be non-zero. Therefore (0, 1, 0) is the point of intersection counted with multiplicity 2 in agreement with the theorem.