Problem Reduction to Parameter Space

Kim, Myung-Soo; Elber, Gershon

doi:10.1007/978-1-4471-0495-7_6

Cited by 15 publications

(15 citation statements)

References 19 publications

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“…The rigidity (12) measures the deviation of the inner products of moving test vectors A(t) q i from constant functions, as it involves integrals of squares of derivatives. Consequently, it represents the change of angles and sizes of test vectors which occurs during the motion of the object.…”

Section: Rigidity Constraintsmentioning

confidence: 99%

Minimizing the Distortion of Affine Spline Motions

Hyun¹,

Jüttler²,

Kim³

2002

Graphical Models

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Section: Rigidity Constraintsmentioning

confidence: 99%

Minimizing the Distortion of Affine Spline Motions

Hyun¹,

Jüttler²,

Kim³

2002

Graphical Models

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“…Techniques for solving a set of polynomial equations are developed and applied to various geometric problems as a primitive tool [Sherbrooke and Patrikalakis 1993;Elber and Kim 2005;Dokken 1985;Grandine et al 2000]. The minimum distance algorithm employed in this paper also operates on the same premise as taken in [Kim and Elber 2000;Patrikalakis and Maekawa 2002].…”

Section: Problem Reduction To Parameter Spacementioning

confidence: 99%

A higher dimensional formulation for robust and interactive distance queries

Seong

Johnson

Cohen

2006

Proceedings of the 2006 ACM Symposium on Solid and Physical Modeling

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We present an efficient and robust algorithm for computing the minimum distance between a point and freeform curve or surface by lifting the problem into a higher dimension. This higher dimensional formulation solves for all query points in the domain simultaneously, therefore providing opportunities to speed computation by applying coherency techniques. In this framework, minimum distance between a point and planar curve is solved using a single polynomial equation in three variables (two variables for a position of the point and one for the curve). This formulation yields twomanifold surfaces as a zero-set in a 3D parameter space. Given a particular query point, the solution space's remaining degrees-offreedom are fixed and we can numerically compute the minimum distance in a very efficient way. We further recast the problem of analyzing the topological structure of the solution space to that of solving two polynomial equations in three variables. This topological information provides an elegant way to efficiently find a global minimum distance solution for spatially coherent queries. Additionally, we extend this approach to a 3D case. We formulate the problem for the surface case using two polynomial equations in five variables. The effectiveness of our approach is demonstrated with several experimental results.

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“…In other works, we presented algorithms that are based on similar reduction to parameter space schemes [25,26,27]. The algorithm for computing Voronoi diagrams that we employ in this paper is based on the same approach as in [19].…”

Section: Introductionmentioning

confidence: 99%

Voronoi diagram computations for planar NURBS curves

Seong

Cohen

Elber

2008

Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling

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We present robust and efficient algorithms for computing Voronoi diagrams of planar freeform curves. Boundaries of the Voronoi diagram consist of portions of the bisector curves between pairs of planar curves. Our scheme is based on computing critical structures of the Voronoi diagrams, such as self-intersections and junction points of bisector curves. Since the geometric objects we consider in this paper are represented as freeform NURBS curves, we were able to reformulate the solution to the problem of computing those critical structures into the zero-set solutions of a system of nonlinear piecewise rational equations in parameter space. We present a new algorithm for computing error-bounded bisector curves using a distance surface constructed from error-bounded offset approximations of planar curves. This error-bounded algorithm is fast and produces bisector curves that are correct both in the topology and the geometry. Once bisectors are computed, both local and global self-intersections of the bisector curves are located and trimmed away by solving a system of three piecewise rational equations in three variables. Further, our method computes junction points at which three or more trimmed bisector curves intersect by transforming them into the solutions to a system of piecewise rational equations in the merged parameter space of the planar curves. The bisectors are trimmed at those self-intersection and global junction points. The Voronoi diagram is then computed from the trimmed bisectors using a pruning algorithm. We demonstrate the effectiveness of our approach with several experimental results.

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Problem Reduction to Parameter Space

Cited by 15 publications

References 19 publications

Minimizing the Distortion of Affine Spline Motions

Minimizing the Distortion of Affine Spline Motions

A higher dimensional formulation for robust and interactive distance queries

Voronoi diagram computations for planar NURBS curves

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