Computational Methods for Algebraic Spline Surfaces: ESF by Tor Dokken, Bert Jüttler

By Tor Dokken, Bert Jüttler

The papers integrated during this quantity offer an summary of the cutting-edge in approximative implicitization and numerous similar subject matters, together with either the theoretical foundation and the present computational techniques. The novel inspiration of approximate implicitization has bolstered the present hyperlink among machine Aided Geometric layout and classical algebraic geometry. there's a transforming into curiosity from researchers and execs 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 study. This quantity will help this trade of principles among many of the groups.

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Proof. Let V be the set of points p ∈ C, which project onto p′ . From the previous definition, we directly deduce that the projection of the tangent cone at p ∈ C is contained in the tangent cone of the projection of p. Thus the tangent cone of C ′ at p′ = (α, β) contains the projection of the tangent cones of the points p ∈ V . Since p ′ is regular, its tangent cone is a line parallel to the y direction. Therefore, the tangent cones of the points p ∈ V are in the plane x − α = 0, parallel to the plane (y, z).

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. The cases for which we have to do a change of coordinates are those where a component of C is in a plane parallel to (y, z) or where a plane parallel to (y, z) is tangent to C in two distinct points.

Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling an A. McKenney, S. Ostrouchov, and D. Sorensen. LAPACK Users’ Guide. SIAM, Philadelphia, 1992. org/lapack/. 3. S. Basu, R. -F. Roy. Algorithms in Real ALgebraic Geometry. SpringerVerlag, Berlin, 2003. ISBN 3-540-00973-6. 4. L. Bus´e, M. Elkadi, and B. Mourrain. Using projection operators in computer aided geometric design. In Topics in Algebraic Geometry and Geometric Modeling,, pages 321–342. Contemporary Mathematics, 2003.

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