Robust set operations on polyhedral solids

Hoffmann, Christoph; Hopcroft, John E.; Karasick, Michael

doi:10.1109/38.41469

Cited by 86 publications

(35 citation statements)

References 8 publications

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“…This clean separation make it possible to implement algorithms without having to worry about the fact that floating-point arithmetic is approximate and can induce errors [HHK89,McC98]. Our implementation relies on a symbolic representation of rational numbers with numerator and denominator of arbitrary sizes.…”

Section: Data Structure Representationmentioning

confidence: 99%

Implicit Real Vector Automata

Boigelot

Brusten

Degbomont

2010

Electron. Proc. Theor. Comput. Sci.

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This thesis introduces a new data structure, the Implicit Real Vector Automaton (IRVA), suited for representing symbolically polyhedra, i.e., regions of n-dimensional space defined by finite Boolean combinations of linear inequalities. IRVA can represent exactly arbitrary convex and non-convex polyhedra, including features such as open and closed boundaries, unconnected parts, and non-manifold components. In addition, they provide efficient procedures for deciding whether a point belongs to a given polyhedron, and determining the polyhedron component (vertex, edge, facet, . . . ) that contains a point.An advantage of IRVA is that they can easily be minimized into a canonical form, which leads to a simple and efficient test for equality between represented polyhedra. Elementary IRVA representing primitive polyhedra, such as linear (in)equations and vector spaces are easily constructed and algorithms have been developed for computing Boolean combinations as well as projections of polyhedra represented by IRVA.These algorithms are illustrated by complete examples of executions as a support for the comprehension of their mechanisms.Another contribution is a first prototype implementation of an IRVA library, containing functions for building and manipulating arbitrary n-dimensional polyhedra. We reinforce the presentation of the implementation by discussing some design choices. Such choices include the use of exact arithmetic.Finally, experimental results are presented and discussed. These experiments pave the way to future adaptations and improvements.

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Section: Data Structure Representationmentioning

confidence: 99%

Implicit Real Vector Automata

Boigelot

Brusten

Degbomont

2010

Electron. Proc. Theor. Comput. Sci.

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“…There are drawbacks however, as high precision routines are needed for all primitive numerical computations, making algorithms highly machine dependent. Furthermore, the required With finite precision the computed value y, of Yo is given by (y, -"de"~-,,)(1 + ,") ( ) [11], and Karasick [12], propose using geometric reasoning and apply it to the problem of polyhedral intersections, however fail to provide a proof of correctness. Sugihara [15] uses geometric reasoning to avoid redundant decisions, which lead to topological inconsistency, in the construction of planar Voronoi diagrams.…”

Section: Robustness Under Finite Precision Arithmeticmentioning

confidence: 99%

Polygon nesting and robustness

Bajaj

Dey

1990

Information Processing Letters

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“…Hopcroft and Karasick [12), and Karasick [15], propose using geometric reasoning and apply it to the problem of polyhedral intersections. Sugihara (22) uses geometric reasoning to avoid redundant decisions and thereby eliminate topological inconsistencies in the construction of planar Voronoi diagrams.…”

Section: Description Of the Algorithmmentioning

confidence: 99%

Convex Decomposition of Polyhedra and Robustness

Bajaj¹,

Dey²

1992

SIAM J. Comput.

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We present a simple algorithm to compute a convex decomposition of a non-convex, non-manifold polyhedron of arbitrary genus (handles). The algorithm takes a non-convex polyhedron with n edges and r notches (features causing non-convexity in the polyhedra) and produces a worst-case optimal O(r 2 ) number of convex polyhedra Si, with U;S; = S, in O(nr 2 ) time and O(nr) space. Recenlly, Chazelle and Patios have given a fast O(n r + r 2 logr) time algorithm to tetrahedraljze a non-convex simple polyhedron. Their algorithm, however, works for a simple polyhedron of genus 0 and with no shells (inner boundaries). The input polyhedron of our algorithm may have arbitrary genus and inner boundaries and may be a non-manifold. We also present an algorithm for the same problem while doing only finite precision a.rithmetic computations.

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Robust set operations on polyhedral solids

Cited by 86 publications

References 8 publications

Implicit Real Vector Automata

Implicit Real Vector Automata

Polygon nesting and robustness

Convex Decomposition of Polyhedra and Robustness

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