By Mohamed Elkadi, Bernard Mourrain, Ragni Piene

Algebraic Geometry offers a powerful conception focusing on the certainty of geometric gadgets outlined algebraically. Geometric Modeling makes use of on a daily basis, with a view to remedy useful and tough difficulties, electronic shapes in keeping with algebraic types. during this publication, we have now accrued articles bridging those components. The disagreement of the various issues of view ends up in a greater research of what the foremost demanding situations are and the way they are often met. We specialise in the subsequent vital periods of difficulties: implicitization, type, and intersection. the combo of illustrative photographs, particular computations and evaluate articles might help the reader to address those matters.

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It is characterized by the simultaneous approximation of sampled point data pi = (xi , yi , zi ), i ∈ I, and estimated unit normals ni at these points. , from a given parametric or procedural description of the surface, then each normal vector ni is estimated from the nearest neighbors of the point pi = (xi , yi , zi ). In addition, a regiongrowing-type algorithm is used for adjusting the orientation of the normal vectors. In order to control the shape of the resulting surface, an additional tension term is optimized.

9. 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.

9 sec. 7 sec. 64 sec. 9 sec. 76 sec. 16 sec. 72 sec. 96 sec. F. Shalaby et al. Piecewise approximate implicitization: experiments using industrial data 47 3) are used. For both algorithms, the implicitization is performed in reasonable time, with reasonable memory usage, and with relatively small error. There are some small diﬀerences between the algorithms. The results of ML and PS are also shown in the same table. Both methods take a parametric surface in terms of a single polynomial patch and a chosen degree of the implicit representation as input.