By Mohamed Elkadi, Bernard Mourrain, Ragni Piene
Algebraic Geometry offers a magnificent conception concentrating on the knowledge of geometric gadgets outlined algebraically. Geometric Modeling makes use of each day, so one can resolve sensible and tough difficulties, electronic shapes according to algebraic types. during this ebook, we now have accumulated articles bridging those parts. The disagreement of the various issues of view leads to a greater research of what the major demanding situations are and the way they are often met. We specialise in the next vital periods of difficulties: implicitization, type, and intersection. the combo of illustrative images, specific computations and assessment articles may help the reader to address those topics.
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Additional info for Algebraic Geometry and Geometric Modeling
The ﬁrst group contains “academic” examples, which were constructed at SINTEF and Linz in order to help developing the algorithms and to display essential features like singularities and self-intersections in a relatively simple setting. The second group consists of industrial examples, provided by CAD vendor think3 (a partner in the GAIA II project). The surfaces are visualized and described in Fig. 1 and Fig. 2. The tables in these ﬁgures give a short description of the various test cases, along with short motivations for choosing these examples.
Both PPL and PPS are able to compute an approximate implicitization for all test cases with reasonable error. In the case of one-patch parametric surfaces, PS (and similarly PPS) is able to reproduce the exact implicitization (within tolerances) if the exact degree is chosen. ML reproduces the exact implicitization if the symbolic integration option is used. Using this option the computation is extremely slow. For our test cases, we used the numerical option. PPL does not reproduce the exact implicit representation, since it approximates not only the points, but also the estimated unit normals.
Zhang. On the validity of implicitization by moving quadrics for rationnal surfaces with no base points. J. Symbolic Computation 29 (2000), 419–440. 10. C. D’Andr´ea. Resultants and moving surfaces. J. of Symbolic Computation 31 (2001), 585–602. 11. D. Eisenbud. Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics 150. Springer-Verlag, New York, 1995. 12. W. Fulton. Intersection theory. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Springer-Verlag, Berlin, 1998.
Algebraic Geometry and Geometric Modeling by Mohamed Elkadi, Bernard Mourrain, Ragni Piene