Weak Approximate Implicitization

Dokken, Tor; Thomassen, Jan B.

doi:10.1109/smi.2006.43

Cited by 9 publications

(22 citation statements)

References 9 publications

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“…The accuracy of the construction of this matrix can be confirmed by checking that the elements sum to 1 2 (see Theorem 2 in [5]). Again, performing an SVD on this matrix and choosing the vector corresponding to the smallest singular value will define an implicit equation that is a candidate for approximation: Although this simple example has no extraneous branches, in order to illustrate the possibility of modelling the shape of the approximation, we include a combined approximation.…”

Section: A First Examplementioning

confidence: 94%

“…In this section we outline the approach to approximate implicitization presented in [2,3], in the context of Bézier triangles. Both the original approach and the so-called weak approach will be described, closely following the procedure given in [5]. We will also look at a numerical approach to the algorithm in greater detail.…”

Section: Approximate Implicitizationmentioning

confidence: 99%

“…The theory behind this approach to approximate implicitization of rational parametric manifolds in R l has been thoroughly developed in [2,3]. A similar approach known as weak approximate implicitization was developed in [5]. The aim of this paper is to present both these methods, in the special case of approximate implicitization of triangular Bézier surfaces.…”

Section: Introductionmentioning

confidence: 99%

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Approximate implicitization of triangular Bézier surfaces

Barrowclough

Dokken

2010

Proceedings of the 26th Spring Conference on Computer Graphics

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Abstract. We discuss how Dokken's methods of approximate implicitization can be applied to triangular Bézier surfaces in both the original and weak forms. The matrices D and M that are fundamental to the respective forms of approximate implicitization are shown to be constructed essentially by repeated multiplication of polynomials and by matrix multiplication. A numerical approach to weak approximate implicitization is also considered and we show that symmetries within this algorithm can be exploited to reduce the computation time of M. Explicit examples are presented to compare the methods and to demonstrate properties of the approximations.

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Section: A First Examplementioning

confidence: 94%

Section: Approximate Implicitizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximate implicitization of triangular Bézier surfaces

Barrowclough

Dokken

2010

Proceedings of the 26th Spring Conference on Computer Graphics

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“…The idea behind approximate implicitization [8][9][10][11][12] is to approximate a rational parametric surface p(s, t), (s, t) ∈ Ω ⊂ IR 2 , Ω being a closed and bounded set, by an algebraic surface q(x, y, z) = 0 of degree m > 0. This differs from exact implicitization, where an exact algebraic representation of the parametric surface is to be found.…”

Section: Approximate Implicitizationmentioning

confidence: 99%

Approximate implicitization and recursive surface intersection algorithms

Dokken

2007

Proc Appl Math and Mech

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Most published work on intersection algorithms for Computer Aided Design (CAD) systems addresses transversal intersections [1], situations where the surface normals of the surfaces intersected are well separated along all intersection curves. For transversal intersections the divide and conquer strategy of recursive subdivision, Sinha's theorem [2] and the convex hull property of NonUniform Rational B-Spline surfaces (NURBS) efficiently identify all intersection branches. However, in singular or near singular intersections, situations where the surfaces are parallel or near parallel in an intersection region, along an intersection curve or in an intersection point, even deep levels of subdivision will frequently not sort out the intersection topology. The paper will focus on the novel approach of Approximate Implicitization to address these challenges.

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“…A valid alternative to exact methods is approximate implicitization; cf. [3,4]. Instead of the exact variety, a low degree approximation is used to represent the shape of the geometric object in a certain region of interest.…”

Section: Introductionmentioning

confidence: 99%

Fast Approximate Implicitization of Envelope Curves Using Chebyshev Polynomials

Barrowclough¹,

Jüttler²,

Schulz³

2012

Latest Advances in Robot Kinematics

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Consider a rational family of planar rational curves in a certain region of interest. We are interested in finding an approximation to the implicit representation of the envelope. Since exact implicitization methods tend to be very costly, we employ an adaptation of approximate implicitization to envelope computation. Moreover, by utilizing an orthogonal basis in the construction process, the computational times can be shortened and the numerical condition improved. We provide an example to illustrate the performance of our approach.

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Weak Approximate Implicitization

Cited by 9 publications

References 9 publications

Approximate implicitization of triangular Bézier surfaces

Approximate implicitization of triangular Bézier surfaces

Approximate implicitization and recursive surface intersection algorithms

Fast Approximate Implicitization of Envelope Curves Using Chebyshev Polynomials

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