Rational parameterizations of curves and surfaces are frequently used in Computer Aided Geometric Design (CAGD) and Algebraic Geometry, where the most common representations are based on the power basis and the Bernstein-Bézier basis. We can concentrate on one of them, since these two representations are closely related by a projective transformation of the parameter domain. By considering rational parameterizations in projective space one can avoid the use of rational functions and work with polynomials instead. The problem of detecting symmetries and equivalences of curves and surfaces attracted substantial attention since it is an essential problem in Pattern Recognition, Computer Graphics and Computer Vision. Knowledge about symmetries helps analyzing pictures and is used for compression and shape completion. Equivalence detection is used to identify a given object with objects in a database. We note here, that the detection of symmetries is a special case of equivalence detection.
It is known, that proper parameterizations of rational curves in reduced form are unique up to bilinear reparameterizations, i.e., projective transformations of the parameter domain. This observation has been used in a series of papers by Alcázar et al. to formulate algorithms for detecting Euclidean equivalences as well as similarities (which include also some scaling) for rational planar and space curves. In this work we generalize this approach in several directions.
In a first step we consider projective and affine equivalences of curves in arbitrary dimensions. Equivalences with respect to the group of projective transformations are the most general of those ones mentioned above and we can treat Euclidean equivalences, similarities and affine equivalences as special cases. Moreover, the freedom of considering arbitrary dimensions is an advantage of our approach. As a second step we state, that for proper, base point free rational surfaces a similar property can be shown, i.e., these representations are unique up to a projective transformation of the parameter domain, which we identify with the projective plane. Furthermore, again we use this insight to detect projective equivalences of surfaces in arbitrary dimension.
We use these observations about rational curves and surfaces to characterize equivalences and symmetries by a polynomial system of equations in the variables describing the linear rational reparameterization. We solve this system using the Gröbner basis implementation of Maple, which is one of the standard computer algebra systems. We provide a substantial number of examples to illustrate our method. Among other results, this allows us to verify known results about the classifications of quadratically parameterized surfaces in a simple way.