Using invariants to extract geometric characteristics of conic sections from rational quadratic parameterizations

Abstract

Extracting the geometric characteristics of conic sections, such as their center, axes and foci, from their defining equations is required for various applications in computer graphics and geometric modeling. Although there exist standard techniques for computing the geometric characteristics for conics in implicit form, in shape modeling applications conic sections are often represented by rational quadratic parameterizations. Here we present closed formulas for computing the geometric characteristics of conics directly from their quadratic parameterizations without resorting to implicitization procedures. Our approach uses the invariants of rational quadratic parameterizations under rational linear reparameterizations. These invariants are also used to give a complete characterization of degenerate conics represented by rational quadratic parameterizations.