Correspondences in Projective Geometry

Extract

A familiar method of proving theorems in elementary projective geometry is to establish by a geometrical construction the existence of a (1, 1) correspondence between the elements of two simply infinite rational systems (such as ranges of points, or pencils of lines) and to conclude that such a correspondence is projective and to use this result to deduce the theorem desired. The argument underlying this reasoning is that the relation between the coordinates x, y of corresponding elements (x), (y) of the two systems is given by an algebraic equation f(x, y) = 0, and that if this relation is such that to each value of x corresponds one value of y and vice versa, then the relation f(x, y) must reduce to a bilinear equation, of the form Axy + Bx + Cy + D = 0, which is easily shown to define a projectivity between the corresponding elements (x), (y).