0/0 Simplifies implicitization

Abstract

A base point of a surface rational parametrization maps to an indeterminate surface point (0/0,0/0,0/0). A quotient resultant becomes indeterminate when it specializes to 0/0. These double anomalies seem hazardous to implicitization. But surprisingly, when they do happen simultaneously, the implicitization result may become simpler. This desirable situation arises when base points are due to a corner-cut parametric monomial support and the corresponding quotient resultant is indeterminate due to the collinearity of corner control vertices. The simplification effect is quite substantial: the implicit polynomial is an a priori known maximal minor from the numerator determinant of the quotient resultant, consequently not only the implicitization determinant is shrunk but division is avoided altogether. © 2007 Elsevier Ltd. All rights reserved.