Robust polyhedral Minkowski sums with GPU implementation

Kyung, Min-Ho; Sacks, Elisha; Milenkovic, Victor

doi:10.1016/j.cad.2015.04.012

Cited by 13 publications

(6 citation statements)

References 20 publications

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“…For that angle range, the boundary of the rotational free space lies between the boundaries of the translational free spaces of the approximations. We can use our fast polyhedral Minkowski sum software [4] to generate approximations of the rotational free space boundary for a set of angle intervals covering the unit circle. We see no reason that sweeping a relevant facet should have fewer identities than sweeping an irrelevant facet, and so fast identity detection should provide the same speedup as observed in Sec.…”

Section: Discussionmentioning

confidence: 99%

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Fast Detection of Degenerate Predicates in Free Space Construction

Milenkovic

Sacks

Butt

2019

Int. J. Comput. Geom. Appl.

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The key to a robust and efficient implementation of a computational geometry algorithm is an efficient algorithm for detecting degenerate predicates. We study degeneracy detection in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to another polyhedron. The structure of the free space is determined by the signs of univariate polynomials, called angle polynomials, whose coefficients are polynomials in the coordinates of the vertices of the polyhedra. Every predicate is expressible as the sign of an angle polynomial f evaluated at a zero t of an angle polynomial g. A predicate is degenerate (the sign is zero) when t is a zero of a common factor of f and g. We present an efficient degeneracy detection algorithm based on a one-time factoring of every possible angle polynomial. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time. ACM Subject Classification Theory of computation → Computational geometry

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Section: Discussionmentioning

confidence: 99%

“…We address identities in four prior works. We [4] compute polyhedral Minkowski sums orders of magnitude faster than Hachenberger by using a convolution algorithm, which has fewer identities, and by detecting identities with special case logic. We [6] compute free spaces of planar parts bounded by circular arcs and line segments.…”

Section: Prior Workmentioning

confidence: 99%

Fast Detection of Degenerate Predicates in Free Space Construction

Milenkovic

Sacks

Butt

2019

Int. J. Comput. Geom. Appl.

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“…In addition, no inner surfaces can be represented here either. Kyung et al [12] describe a robust convolutional algorithm to calculate the Minkowski sum of two polyhedra, that finds and removes intersecting facets using kd-trees and achieves high accuracy.…”

Section: Related Workmentioning

confidence: 99%

Variable-Radius Offset Surface Approximation on the GPU

Woerl¹,

Schoemer²,

Schwanecke³

2020

Computer Science Research Notes

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Variable-radius offset surfaces find applications in various fields, such as variable brush strokes in 2D and 3D sketching and geometric modeling tools. In forensic facial reconstruction the skin surface can be inferred from a given skull by computing a variable-radius offset surface of the skull surface. Thereby, the skull is represented as a two-manifold triangle mesh and the facial soft tissue thickness is specified for each vertex of the mesh. We present a method to interactively visualize the wanted skin surface by rendering the variable-radius offset surfaces of all triangles of the skull mesh. We have also developed a special shader program which is able to generate a discretized volumetric representation that can be transformed into a skin mesh. In addition, we show the usefulness of our method to calculate classical Minkowski sums and demonstrate its use for packing problems.

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“…5). We construct the Minkowski sums robustly using our prior algorithm [3], but without perturbing the input. The output vertex coordinates are ratios of degree 7 and degree 6 polynomials in the input coordinates, hence have about 800 bit precision.…”

Section: Minkowski Sumsmentioning

confidence: 99%

“…Mesh construction using ECG is common in computational geometry research and plays a growing role in applications. We use ECG to implement Booleans, linear transformations, offsets, and Minkowski sums [2,3]; the CGAL library [4] provides many ECG implementations.…”

Section: Introductionmentioning

confidence: 99%

Geometric rounding and feature separation in meshes

Milenkovic

Sacks

2019

Computer-Aided Design

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Geometric rounding of a mesh is the task of approximating its vertex coordinates by floating point numbers while preserving mesh structure. Geometric rounding allows algorithms of computational geometry to interface with numerical algorithms. We present a practical geometric rounding algorithm for 3D triangle meshes that preserves the topology of the mesh. The basis of the algorithm is a novel strategy: 1) modify the mesh to achieve a feature separation that prevents topology changes when the coordinates change by the rounding unit; and 2) round each vertex coordinate to the closest floating point number. Feature separation is also useful on its own, for example for satisfying minimum separation rules in CAD models. We demonstrate a robust, accurate implementation.

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Robust polyhedral Minkowski sums with GPU implementation

Cited by 13 publications

References 20 publications

Fast Detection of Degenerate Predicates in Free Space Construction

Fast Detection of Degenerate Predicates in Free Space Construction

Variable-Radius Offset Surface Approximation on the GPU

Geometric rounding and feature separation in meshes

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