New PDF release: Computational Methods for Algebraic Spline Surfaces. ESF

By Tor Dokken, Bert Jüttler

ISBN-10: 3642062334

ISBN-13: 9783642062339

The papers incorporated during this quantity supply an summary of the state-of-the-art in approximative implicitization and diverse similar subject matters, together with either the theoretical foundation and the present computational techniques.The novel proposal of approximate implicitization has bolstered the present hyperlink among computing device Aided Geometric layout and classical algebraic geometry. there's a starting to be curiosity from researchers and execs either in CAGD and Algebraic Geometry, to fulfill andcombine wisdom and ideas,with the purpose to enhance the fixing of industrial-type demanding situations, in addition to to start up new instructions for simple examine. This quantity will aid this alternate of rules among many of the groups.

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Additional resources for Computational Methods for Algebraic Spline Surfaces. ESF Exploratory Workshop

Example text

Proof. We denote by L the regular section at x = α of C and by L the next section, at x = α . Let p = (α, β, γ) ∈ L, q = (α, β, δ) ∈ L with γ = δ. They are connected by C respectively to p = (α , β , γ ), q = (α , , δ ) ∈ L . Assume that β = . Then there are two arcs of the projection C , connecting (α, β) to (α , β ) and to (α , ), above [α, α ]. This implies that there exists a point r ∈ C with x(r) ∈ [α, α [ belonging to 3 branches. Such a point cannot be regular, in contradiction with the fact that C is smooth above [α, α [.

An ) ∈ Cn then p 2 is defined as |a1 |2 + · · · + |an |2 . We start with the notion of –point on an algebraic affine plane curve. As we have mentioned in the introduction, the notion of –point is quite intuitive and, for the curve case, it essentially consists in points such that when substituted in the implicit equation of the curve one gets values of small modulus. Nevertheless, for our purposes and taking into account that we will be working in the frame of algebraic geometry, we are also interested in introducing the additional notion of –singularity.

Z q−1 P, . . , Q, z Q, . . , z p−1 Q = dim D, z D, . . , z p+q−1−δ D , we deduce that corank(Syl(P ,Q)) = δ. 30 G. Gatellier et al. e that admit a finite number of (complex) solutions. Different approaches exist to solve such systems [8]. We focus on the algebraic approach that transforms the resolution problem into linear algebra problems. Here are some notations: R = R[x, y, z], f1 = 0, . . , fm = 0 with fi ∈ R, the equations we want to solve, I = (f1 , . . , fm ) is the ideal generated by these polynomials, A = R/I the quotient algebra.

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Computational Methods for Algebraic Spline Surfaces. ESF Exploratory Workshop by Tor Dokken, Bert Jüttler


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