Change of representation
In many geometric algorithms, combining parametric and implicit representations of the geometric objects processed improves the quality of the results and computational performance. Algorithms that convert from a parametric to an implicit representation (implicitization), and algorithms which convert from an implicit representation to a parametric (parameterization) are of great value within geometry processing.
For curves such as straight lines, circles, ellipses and hyperbolas both rational parametric and implicit representations exist. The same is true for planes, spheres, ellipsoids, hyperboloids, etc. However, for curves and surfaces that have an implicit description of total degree higher than two, we do not have this attractive duality.
The GAIA II project has addressed two approaches to implicitization:
The GAIA II project has thus addressed two approaches to parameterization: