The geometric interpretation of inversion formulae for rational plane curves

Abstract

Given a faithful parameterization P(t) of a rational plane curve, an inversion formula t = f(x,y) gives the parameter value corresponding to a point (x,y) on the curve, where f is a rational function in x and y. We investigate the relationship between a point (x*,y*) not on the curve and the corresponding point P(t*) on the curve, where t* = f(x*,y*). It is shown that for a rational quadratic plane curve, P(t*) is the projection of (x*,y*) from a point which may be any point on the curve; for a rational cubic plane curve, P(t*) is the projection of (x*,y*) from the double point of the curve. Applications of these results are discussed and a generalized result is proved for rational plane curves of higher degree. © 1995.