The geometric interpretation of inversion formulae for rational plane curves Original Research Article

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.