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Prototype approximate implicitization

Wider acceptance. The prototype for approximate implicitization is a scientific result introducing a numeric approach to implicitization of rational parametric surface instead of the traditional approaches requiring symbolic computations and exact arithmetic. The general idea of approximate implicitization was introduced in the Dr. Philos thesis from 1997 of Tor Dokken (Coordinator of the GAIa II project) and has been further elaborated in the GAIA II project. Approximate implicitization is at the end of the GAIA II project beginning to be accepted as a well documented approach also within applied real algebraic geometry. Achievements during GAIA II.
At the start of the GAIA II project approximate implicitization was dependent on the use of the Bernstein basis for the rational parametric manifold to be approximated.

This dependence on the Bernstein basis forced the central matrices to be built to use multiplication of polynomials. During the project the dependence on the Bernstein basis for practical implementation has been removed, and the approach can now be built on numeric integration, that give smaller matrices. At the end of the project we have understood why the use of the Bernstein basis give matrices with well distributed singular values, although we have not have time to publish this yet. It also seems that the theory behind approximate implicitization can give new insight into the convergence of rational parametric approximation. However, more research is needed to final establish this.

Software implemented:
A number of software prototypes related to approximate implicitization have been developed during the project:
- The traditional approach with multiplication of polynomials.
- A sampling based approach.
- An approach for piecewise approximate implicitization.

These best working of these prototypes will be made available as open source software under the GNU GPL license.

Dissemination potential:
The result will be disseminated through:
- Conference presentations and paper, a number of these have already taken place, more are planned.
- Opens Source GNU GPL License for non-commercial use.
- Commercial licensing.

Use of the result and expected benefits:
We have already demonstrated the usefulness of the result with respect to separating near parallel curve and surfaces in CAD intersection algorithms, and the potential for supporting detection of self-intersection in CAD-curves and surfaces.

There is, however, a not exploited potential with respect to shape design, where first a shape can be sketched using traditional CAD-techniques, the approximate implicitization can be used to catch the design intent and for modification of shape. This approach can help us solve the dependence of CAD-surface design on planar curve, and also help us design and modify 3D freeform curves.

By making the implementation of the approach available under the GNU GPL license we hope to further promote the approach for academic studies as well as for industrial testing.

The documentation and software of the software prototype will be made avaiable on the web-link:

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