The concept of “Approximate Implicitization” was introduced in the Dr. Philos. dissertation of Tor Dokken [1] from 1997, this was made more readily available in 2001 [2] and in 2003 [3]. In GAIA II we have addressed extensions to the theory and new approaches to approximate implicitization to improve flexibility, performance and better control the behavior of the implicit curves and surfaces resulting.

Approximate implicitization by factorization

• The approach is numerically stable and reformulates implicitization to finding the smaller singular value problem of a matrix of real numbers.
•  The approach can be used as an exact implicitization method if the proper degree is chosen for the unknown implicit and exact arithmetic is used.
• The approach has high convergence rates and is numerical stable.
• Strategies for selecting solutions with a desired gradient behavior are supplied, either for encouraging vanishing gradients or avoiding vanishing gradients.
• The approach works both for rational parametric curves and surfaces, and for procedural surfaces.
• Experiments with piecewise algebraic curves and surfaces have produced implicit curves and surfaces that have more vanishing gradients than is desirable.  Possibly estimating gradients will improve this challenge.

Approximate implicitization by point sampling & normal estimates

• This approach is constructive in nature as it estimates gradients of the implicit representation to ensure that gradients do not vanish when not desired.
• The approach produce good implicit curves and surfaces
• Vanish of gradients in not desired regions is minimal
• The method works well for approximation by piecewise implicit curves and surfaces