Matrix-based implicit representations of rational algebraic curves and applications

Given a parameterization of an algebraic rational curve in a projective space of arbitrary dimension, we introduce and study a new implicit representation of this curve which consists in the locus where the rank of a single matrix drops. Then, we illustrate the advantages of this representation by addressing several important problems of Computer Aided Geometric Design: the point-on-curve and inversion problems, the computation of singularities and the calculation of the intersection between two rational curves.

Research highlights
► Given a parameterized algebraic curve, a new implicit representation, based on a single matrix, is introduced.
► With this new representation, a given point belongs to the curve if and only if the rank of a single matrix drops at this point.
► Computing a Smith form of a matrix built from this new implicit representation of the curve, its singular locus can be determined.
► An algorithm to solve the curve/curve intersection problem at the level of matrices is proposed.