Efficient Groebner walk conversion for implicitization of geometric objects - Tran, Quoc-Nam (Computer aided geometric design, 2004)
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.