Journal of Symbolic Computation
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
The structure of polynomial ideals and Grobner bases
SIAM Journal on Computing
Equations for the projective closure and effective Nullstellensatz
Discrete Applied Mathematics - Special volume on applied algebra, algebraic algorithms, and error-correcting codes
Algorithm for implicitizing rational parametric surfaces
Computer Aided Geometric Design
Implicit Curves and Surfaces in CAGD
IEEE Computer Graphics and Applications - Special issue on computer-aided geometric design
Implicitization using moving curves and surfaces
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
Implicitization of rational parametric surfaces
Journal of Symbolic Computation
Hilbert functions and the Buchberger algorithm
Journal of Symbolic Computation
Converting bases with the Gröbner walk
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
A fast algorithm for Gröbner basis conversion and its applications
Journal of Symbolic Computation - Special issue on applications of the Gröbner basis method
Term Orderings on the Polynominal Ring
EUROCAL '85 Research Contributions from the European Conference on Computer Algebra-Volume 2
Revisiting the µ-basis of a rational ruled surface
Journal of Symbolic Computation
IEEE Computer Graphics and Applications
The midpoint locus of a triangle in a corner
ADG'10 Proceedings of the 8th international conference on Automated Deduction in Geometry
| Hi-index | 0.00 |
In this paper, the author uses recent theoretical results from the method of Groebner bases to improve the efficiency of algorithms for implicitization. The method of Groebner bases has some important advantages, namely it is reliable and it can solve the implicitization problem in full generality. The main result of this paper is that we can significantly improve the efficiency of implicitization algorithms using the deterministic Groebner walk conversion while maintaining the reliability of the algorithms. More precisely, the calculation of the implicit equations will be partitioned into several smaller computations following a path in the Groebner fan of the ideal generated by the system of equations. This method works with ideals of zero-dimension as well as positive dimension. The author uses a deterministic method to vary the weight vectors in order to ensure that the computation involves polynomials with just a few terms. A new concept of ideal-specified term orders for elimination is introduced to further improve the efficiency. As the result, the improved algorithms overcome the bottle-neck of the traditional implicitization algorithms by avoiding unnecessary zero-reductions and coefficient swell. Furthermore, the improved algorithms are able to avoid many unnecessary walking steps during the calculation of the implicit equations. Several test-suites such as the Newell's teapot are used to test the new approach. The average performance is many times faster than traditional Groebner basis based algorithms for implicitization.