Geometric constraint solver using multivariate rational spline functions

Elber, Gershon; Kim, Myung-Soo

doi:10.1145/376957.376958

Cited by 162 publications

(141 citation statements)

References 16 publications

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“…It can be robustly and efficiently computed using a rational constraint solver [10,33]. The robustness is achieved by bounding the subdivided implicit surface I with the corresponding hyper-tangent cone [10], an extension of the bounding tangent cones for explicit plane curves and explicit surfaces [31,32]. Let us recall that the implicit 2-manifold I in R 5 {s,ŝ,t} is the locus of intersection points of the two deforming surfaces, over the whole time period.…”

Section: Detection Of Transition Eventsmentioning

confidence: 99%

See 1 more Smart Citation

Theoretically Based Robust Algorithms for Tracking Intersection Curves of Two Deforming Parametric Surfaces

Chen

Riesenfeld

Cohen

et al. 2006

Geometric Modeling and Processing - GMP 2006

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Abstract. This paper applies singularity theory of mappings of surfaces to 3-space and the generic transitions occurring in their deformations to develop algorithms for continuously and robustly tracking the intersection curves of two deforming parametric spline surfaces, when the deformation is represented as a family of generalized offset surfaces. The set of intersection curves of 2 deforming surfaces over all time is formulated as an implicit 2-manifold I in the augmented (by time domain) parametric space R 5 . Hyper-planes corresponding to some fixed time instants may touch I at some isolated transition points, which delineate transition events, i.e., the topological changes to the intersection curves. These transition points are the 0-dimensional solution to a rational system of 5 constraints in 5 variables, and can be computed efficiently and robustly with a rational constraint solver using subdivision and hyper-tangent bounding cones. The actual transition events are computed by contouring the local osculating paraboloids. Away from any transition points, the intersection curves do not change topology and evolve according to a simple evolution vector field that is constructed in the euclidean space in which the surfaces are embedded.

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Section: Detection Of Transition Eventsmentioning

confidence: 99%

“…In the algorithm, u 1 and u 2 are the two asymptotic directions, which can be solved (for u) from the equation II(u, u) = 0 using the second fundamental form in Eq. (10). The other pair of contours, with the opposite height value, can be sampled similarly, with one of the asymptotic directions reversed.…”

Section: Compute Transition Eventsmentioning

confidence: 99%

Theoretically Based Robust Algorithms for Tracking Intersection Curves of Two Deforming Parametric Surfaces

Chen

Riesenfeld

Cohen

et al. 2006

Geometric Modeling and Processing - GMP 2006

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show abstract

“…Subdivision-based algorithms are one of the most commonly used algorithms in many fields, from computational geometry and graphics to solving of polynomials and mathematical programming [22,18,28,2,23,39,16,1].…”

Section: Introductionmentioning

confidence: 99%

The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

Burr

Gao

Tsigaridas

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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“…In [SP93], the author used tensor product version of Bernstein basis and integrated domain reduction techniques to speed up the convergence and reduce the number of subdivisions. In [EK01], the emphasis is put on the subdivision process, and stopping criterion based on the normal cone to the surface patch. In [MP05], this approach has been improved by introducing pre-conditioning and univariatesolver steps.…”

Section: Introductionmentioning

confidence: 99%

Subdivision Methods for the Topology of 2d and 3d Implicit Curves

Chen¹,

Mourrain²,

Pavone³

2008

Geometric Modeling and Algebraic Geometry

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Abstract. In this paper, we describe a subdivision method for handling algebraic implicit curves in 2d and 3d. We use the representation of polynomials in the Bernstein basis associated with a given box, to check if the topology of the curve is determined inside this box, from its points on the border of the box. Subdivision solvers are used for computing these points on the faces of the box, and segments joining these points are deduced to get a graph isotopic to the curve. Using envelop of polynomials, we show how this method allow to handle efficiently and accurately implicit curves with large coefficients. We report on implementation aspects and experimentations on 2d curves such as ridge curves or self intersection curves of parameterized surfaces, and on silhouette curves of implicit surfaces, showing the interesting practical behavior of this approach.

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Geometric constraint solver using multivariate rational spline functions

Cited by 162 publications

References 16 publications

Theoretically Based Robust Algorithms for Tracking Intersection Curves of Two Deforming Parametric Surfaces

Theoretically Based Robust Algorithms for Tracking Intersection Curves of Two Deforming Parametric Surfaces

The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

Subdivision Methods for the Topology of 2d and 3d Implicit Curves

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