By Mohamed Elkadi, Bernard Mourrain, Ragni Piene

Algebraic Geometry offers a magnificent idea concentrating on the knowledge of geometric gadgets outlined algebraically. Geometric Modeling makes use of on a daily basis, so one can resolve functional and hard difficulties, electronic shapes according to algebraic types. during this ebook, now we have accrued articles bridging those components. The war of words of the various issues of view ends up in a greater research of what the major demanding situations are and the way they are often met. We specialize in the subsequent very important sessions of difficulties: implicitization, category, and intersection. the combo of illustrative photographs, particular computations and evaluation articles may help the reader to deal with those topics.

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Shalaby et al. Quartic surface (PPL) Self pipe(PPL) Quartic surface (PPS) Cut–away view of Self pipe (PPL) Self pipe (PPS) Fig. 3. Results (PPL, PPS) • In the case of spline surfaces, the notion of an exact implicitization does not make much sense. In the case of one-patch parametric surface, PS is the fastest and the most accurate method. • For both PPL and PPS one may increase the number of segments and use a low degree, while maintaining the same level of accuracy. A trivariate tensor-product spline function with k inner knots in each direction has (k + 1)3 cells/segments and (k + d + 1)3 scalar coeﬃcients.

NURBS surfaces – as input, and piecewise polynomial implicit functions as output. In addition to studying the feasibility, the two methods are qualitatively compared to each other. Some of the test surfaces are given in terms of a single polynomial patch, and for this data it is possible to apply the implicitize routine implemented in MAPLE (‘ML’) and the previously studied single polynomial implicitization algorithm from SINTEF (‘PS’) [20]. For these test cases we have been able to compare all four methods (PPL, PPS, ML, PS), and thereby study the eﬀects of using implicit functions that are piecewise polynomial.

Shalaby et al. Fig. 4. Reproducing singularities vs. avoiding unwanted branches. Acknowledgment. This research has been supported by the European Commission through project IST-2001-35512 ‘Intersection algorithms for geometry based IT-applications using approximate algebraic methods’ (GAIA II). References 1. : On local implicit approximation and its applications. ACM Trans. Graphics 8, 4:298–324, (1989) 2. : Numerical implicitization of parametric hypersurfaces with linear algebra. In: AISC’2000 Proceedings, Springer, LNAI 1930.