# Algebraic Geometry and Geometric Modeling by Mohamed Elkadi, Bernard Mourrain, Ragni Piene

By Mohamed Elkadi, Bernard Mourrain, Ragni Piene

Algebraic Geometry offers a magnificent thought focusing on the knowledge of geometric gadgets outlined algebraically. Geometric Modeling makes use of each day, to be able to resolve useful and tough difficulties, electronic shapes in keeping with algebraic versions. during this booklet, we now have accrued articles bridging those parts. The war of words of the various issues of view ends up in a greater research of what the most important demanding situations are and the way they are often met. We specialize in the next very important periods of difficulties: implicitization, category, and intersection. the mix of illustrative photographs, particular computations and evaluate articles might help the reader to deal with those topics.

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The “academic” surfaces were constructed in the course of testing and developing the algorithms, while the industrial surfaces were supplied by the CAD vendor think3, which is a partner in the GAIA II project. We follow a procedure similar to [20], where benchmarking for implicitization by polynomials has been presented. The input was a single parametric patch while the output was a single polynomial implicit function. In this paper, we have generalized this to the case of piecewise polynomials.

2, which addresses the conﬂict between pushing away unwanted branches and avoiding singularities. Due to the compact support of the B-splines, the implementation is relatively fast because of the sparsity of the resulting linear system of equations. Consequently, even complicated singular surfaces can be implicitized. In this case, it is an important issue to create a consistent orientation of the (estimated) normals ni , and this can be achieved by a propagation technique [14]: First, an initial approximation is computed using only information from one part of the data, where a consistent orientation could be created without ambiguities.

As with the ‘nested nodal surface’ the correct implicit degree is reduced, in this case to 4. (7, 7) High degree oscillating patch without singularities. Surface 7x7 The theoretical algebraic degree of this surface is 2 × 7 × 7 = 98. (6, 5) Self-intersecting patch. A self-intersecting curve is Surface 6x5 moved in space and at the same time bent. The theoretical algebraic degree of this surface is 2 × 6 × 5 = 60. Fig. 1. F. Shalaby et al. Self ucurves Self pipe Surface Self ucurves Simplesweep Self sweep Self pipe Self proport Simplesweep Self sweep Self proport Degree Description (4, 2) Surface with a closed self-intersection curve.