On Local Implicit Approximation and Its Applications

A method is proposed for computing an implicit approximant at a point to a parametric curve or surface. The method works for both polynomially and rationally parameterized curves and surfaces and achieves an order of contact that can be prescribed. In the case of nonsingular curve points, the approximant must be irreducible, but in the surface case additional safeguards are incorporated into the algorithm to ensure irreducibility. The method also yields meaningful results at most singularities. In principle, the method is capable of exact implicitization and has a theoretical relationship with certain resultant-based elimination methods.