Projective and affine symmetries and equivalences of rational and polynomial surfaces

Abstract

It is known, that proper parameterizations of rational curves in reduced form are unique up to bilinear reparameterizations, i.e., projective transformations of its parameter domain. This observation has been used in a series of papers by Alcázar et al. to formulate algorithms for detecting Euclidean equivalences and symmetries as well as similarities. We generalize this approach to projective equivalences of rationally parametrized surfaces. More precisely, we observe that a birational base-point free parameterization of a surface is unique up to projective transformations of the domain. Furthermore, we use this insight to find all projective equivalences between two given surfaces. In particular, we formulate a polynomial system of equations whose solutions specify the projective equivalences, i.e., the reparameterizations associated with them. Furthermore, we investigate how this system simplifies for the special case of affine equivalences for polynomial surfaces and how we can use our method to detect projective symmetries of surfaces. This method can be used for classifying the generic cases of quadratic surfaces.