Using projection operators in computer aided geometric design

Abstract

We give an overview of resultant theory and some of its applications in computer aided geometric design. First, we mention different formulations of resultants, including the projective resultant, the toric resultant, and the residual resultants. In the second part we illustrate these tools, and others projection operators, on typical problems as surface implicitization, inversion, intersection, and detection of singularities of a parameterized surface.ki j=0 c i,j ψ i,j (x) = 0 for i = 0, . . . , n. In other words, c is the first projection of the point (c, x) in the incidence variety W X = {(c, x) ∈ P k0 × · · · × P kn × X : f i (x) = 0, i = 0, . . . , n}.

Resultants and Discriminants for Bivariate Tensor-Product Polynomials

“…• adjoint multiplication maps M * h 2 : S * 2 (a2 + b2) → S * 2 (a2) with λ → h2 · λ, expressing the action of a polynomial h2 ∈ S2(b2), as defined in (10).…”

“…We refer the reader to [32] as a basic reference, to [13] for a nice introduction to the theory of resultants, to [10] for its application in Computer-aided Geometric Design (CAGD), Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored.…”

Intersection and self-intersection of surfaces by means of Bezoutian matrices

“…. , f n (c, x) [5,Theorem 3.4] (see also [7,Theorem 3.7] for general statements). Therefore, if ∆(c) stands for the determinant of a maximal minor of the Bezoutian matrix of f 0 (c, x), .…”

“…These operations still need to be improved and computer algebra techniques can help. Among them, generalized resultant (for instance sparse resultant) techniques have been successfully employed [18,19,7].…”

“…Several strategies have been developed to address these problems: either via multivariate resultants [7], or via special case study [14], or via approximate implicitization [9,26], or via numerical methods [13].…”