The Algebra and Geometry of Curve and Surface Inversion

An inversion equation takes the Cartesian coordinates of a point on a parametric curve or surface and returns the parameter value(s) of that point. A 2-D curve inversion equation has the form t = f(x, y)/g(x, y). This paper shows that practical insight into inversion can be obtained by studying the geometry of the implicit curves f(x, y) = 0 and g(x, y) = 0. For example, the relationship between the singular locus of the parametric curve and the lowest possible degree of an inversion equation can be understood in this way. Also, insight is given into what parameter value will be returned if an inversion equation is fed the Cartesian coordinates of a point that does not lie on the curve. The standard method of devising curve and surface inversion equations is as a by-product of the implicitization process. This paper presents a new method for finding inversion equations, which allows us to create new inversion equations that have attractive properties. For example, we can create an inversion equation that, to first order approximation, will return the parameter value of the nearest point on the curve if given a point that does not lie precisely on a parametric curve.