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.

• A rational parametric curve of degree n has an implicit representation of total degree n.
• A rational parametric surface of degrees (n₁,n₂) has an implicit representation of total degree 2n₁n_2.
• For implicit curves and surfaces of total degree two or lower, there are always rational parametric representations
• For implicit curves and surfaces with a total degree higher than two, sometimes a rational parameterization exists, sometimes a rational parameterization does not exist.

Implicitization

The GAIA II project has addressed two approaches to implicitization:

• Exact Implicitization, to be used when mathematically exact implicit representation is needed, and the use of exact arithmetic is necessary.
• Approximate implicitization that uses floating point arithmetic and aims at giving a “good” implicit approximation to the parametric curve or surface in a “region of interest”. The approaches addressed allow for approximation with lower total degrees than the exact implicit degrees, allowing for use of piecewise implicit representations, and approximate implicitization of procedural parametric curves and surfaces (curves and surfaces where only evaluator for points and (partial) derivatives are available).

Parameterization

The GAIA II project has thus addressed two approaches to parameterization:

• Looking into monoid curves and surfaces
• Looking at approximate parameterization