By Tor Dokken, Bert Jüttler
The papers incorporated during this quantity supply an summary of the cutting-edge in approximative implicitization and diverse comparable themes, together with either the theoretical foundation and the prevailing computational techniques. The novel thought of approximate implicitization has reinforced the prevailing hyperlink among desktop Aided Geometric layout and classical algebraic geometry. there's a transforming into curiosity from researchers and pros either in CAGD and Algebraic Geometry, to satisfy and combine wisdom and ideas, with the purpose to enhance the fixing of industrial-type demanding situations, in addition to to begin new instructions for uncomplicated examine. This quantity will help this alternate of principles among a number of the groups.
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Extra resources for Computational Methods for Algebraic Spline Surfaces: ESF Exploratory Workshop
It reduces to computing the projective resultant of these two homogeneous polynomials. Since the number of asymptotic directions of C is finite, by a generic linear change of variables, we can avoid the cases where C has an asymptotic direction parallel to the (y, z) plane. Next, we compute the x-critical points of C by solving the system (1), using algorithm 7. This computation allows us to check that the system is zero-dimensional and that the x-coordinate of the real solutions are distinct. If this is not the case, we perform a generic change of coordinates.
Let us assume that αi ∈ Σ and αi+1 ∈ Σ (the treatment of the other possibility being symmetric). To simplify the notations, let L = Li ⊂ C and L′ = Li+1 ⊂ C. By the genericity assumption, L′ contains at most one x-critical point c of C. Since αi ∈ Σ, each point in L is regular. Moreover, by construction, the arcs of C above Topology of Three-Dimensional Algebraic Curves 37 ]αi , αi+1 [ have no x-critical point. Since there is no asymptotic direction of C in the (y, z)-direction, by the implicit function theorem, a (regular) point p ∈ L is connected by an arc of C, to a point of L′ .
A new criterion for normal form algorithms. In M. Fossorier, H. Imai, Shu Lin, and A. Poli, editors, Proc. AAECC, vol. 1719 of LNCS, pages 430–443. Springer, Berlin, 1999. 18. J. Owen and A. Rockwood. Intersection of general implicit surfaces. In Geometric Modeling: Algorithms and New Trends, pages 335–345. SIAM, Philadelphia, 1987. 19. M. P. Patrikalakis and T. Maekawa. Shape Interrogation for Computer Aided Design and Manufacturing. Springer Verlag, 2002. 20. Ph. Tr´ebuchet. Vers une r´esolution stable et rapide des e´ quations alg´ebriques.
Computational Methods for Algebraic Spline Surfaces: ESF Exploratory Workshop by Tor Dokken, Bert Jüttler