By Mohamed Elkadi, Bernard Mourrain, Ragni Piene
Algebraic Geometry presents a magnificent conception concentrating on the certainty of geometric gadgets outlined algebraically. Geometric Modeling makes use of on a daily basis, to be able to clear up functional and tough difficulties, electronic shapes in accordance with algebraic types. during this ebook, we now have accrued articles bridging those components. The disagreement of the various issues of view leads to a greater research of what the most important demanding situations are and the way they are often met. We specialize in the subsequent vital sessions of difficulties: implicitization, class, and intersection. the combo of illustrative photos, particular computations and assessment articles may help the reader to deal with those topics.
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Extra info for Algebraic Geometry and Geometric Modeling
D. Cox, R. Goldman, M. 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.
It consists of 1824 bicubic patches connected with C 2 continuity. Fig. 2. Industrial test surfaces (data courtesy of think3) Piecewise approximate implicitization: experiments using industrial data 45 The quoted time is an approximation of the processor time used by the program, and similarly for the memory. These quantities, in the case of PPL, PPS and PS, were measured with standard Linux tools (time and top). In the case of ML the ‘proﬁle’ procedure of MAPLE was used. For the error measurements, we consider a set of points pi on the parametric surface and compute their Euclidean distances with respect to the implicit surface, using a Newton-like iteration procedure.
In contrast, PS and ML can be applied only to parametrically deﬁned rational surfaces (patches of NURBS). 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.