Voronoi diagram for multiply-connected polygonal domains I: Algorithm

This paper describes an efficient shape representation framework for planar shapes using Voronoi skeletons. This paper makes the following significant contributions. First a new algorithm for the construction of the Voronoi diagram of a polygon with holes is described. The main features of this algorithm are its robustness in handling the standard degenerate cases (colinearity of more than two points; co-circularity of more than three points), and its ease of implementation. It also features a robust numerical scheme to compute non-linear parabolic edges that avoids having to solve equations of degree greater than two. The algorithm has been fully implemented and tested on a variety of test inputs.Second, the Voronoi diagram of a polygon is used to derive accurate and robust skeletons for planar shapes. The shape representation scheme using Voronoi skeletons possesses the important properties of connectivity as well as Euclidean metrics. Redundant skeletal edges are deleted in apruning step which guarantees that connectivity of the skeleton will be preserved. The resultant representation is stable with respect to being invariant to perturbations along the boundary of the shape. A number of examples of shapes with and without holes are presented to demonstrate the features of this approach.

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“…We demonstrate its use in deriving efficient shape representation schemes. Other applications include mesh generation [38].…”

Section: Discussionmentioning

confidence: 99%

Voronoi diagrams of polygons: A framework for shape representation

Mayya

Rajan

1996

J Math Imaging Vis

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“…The medial axis transformation can be easily extracted by removing the Voronoi edges connecting to concave vertices of the polygon. Srinivasan et al [31] extends Lee's algorithm to computing a generalized Voronoi diagram for polygons with holes. Choi [32] presents an MAT approximation algorithm in the planar domain via domain decomposition.…”

Section: Methods To Compute the Matmentioning

confidence: 99%

A practical path planning methodology for wire and arc additive manufacturing of thin-walled structures

Ding

Pan

Cuiuri

et al. 2015

Robotics and Computer-Integrated Manufacturing

249

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This paper presents a novel methodology to generate deposition paths for wire and arc additive manufacturing (WAAM). The medial axis transformation (MAT), which represents the skeleton of a given geometry, is firstly extracted to understand the geometry. Then a deposition path that is based on the MAT is efficiently generated. The resulting MAT-based path is able to entirely fill any given cross-sectional geometry without gaps. With the variation of step-over distance, material efficiency alters accordingly for both solid and thinwalled structures. It is found that thin-walled structures are more sensitive to step-over distance in terms of material efficiency. The optimal step-over distance corresponding to the maximum material efficiency can be achieved for various geometries, allowing the optimization of the deposition parameters. Five case studies of complex models including solid and thin-walled structures are used to test the developed methodology. Experimental comparison between the proposed MAT-based path patterns and the traditional contour path patterns demonstrate significant improved performance in terms of gap-free cross-sections. The proposed path planning strategy is shown to be particularly beneficial for WAAM of thin-walled structures. Abstract: This paper presents a novel methodology to generate deposition paths for wire and arc additive manufacturing (WAAM). The medial axis transformation (MAT), which represents the skeleton of a given geometry, is firstly extracted to understand the geometry. Then a deposition path that is based on the MAT is efficiently generated. The resulting MATbased path is able to entirely fill any given cross-sectional geometry without gaps. With the variation of step-over distance, material efficiency alters accordingly for both solid and thinwalled structures. It is found that thin-walled structures are more sensitive to step-over distance in terms of material efficiency. The optimal step-over distance corresponding to the maximum material efficiency can be achieved for various geometries, allowing the optimization of the deposition parameters. Five case studies of complex models including solid and thin-walled structures are used to test the developed methodology. Experimental comparison between the proposed MAT-based path patterns and the traditional contour path patterns demonstrate significant improved performance in terms of gap-free cross-sections. The proposed path planning strategy is shown to be particularly beneficial for WAAM of thin-walled structures.

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“…Although the edges of an ordinary point Voronoi diagram are linear, higher-order curve sites produce complex boundaries of the diagram composed of high-degree algebraic curves and surfaces. Due to these complexities, algorithms for computing generalized Voronoi diagrams have been developed primarily for linear or circular arc inputs [18,28,30,29]. To fit those algorithms, rational curved inputs typically are preprocessed into linear and circular segments for computing their Voronoi diagram [15,17], which results in a Voronoi diagram that is an approximation in both topology and geometry.…”

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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Voronoi diagram for multiply-connected polygonal domains I: Algorithm

Abstract: Voronoi diagrams of multiply-connected polygonal domains (polygons with holes) can be of use in computer-aided design. We describe a simple algorithm that computes such Voronoi diagrams in OiNilog^N + H)) time, where N is the number of edges and H is the number of holes.

Cited by 84 publications

References 7 publications

Voronoi diagrams of polygons: A framework for shape representation

Voronoi diagrams of polygons: A framework for shape representation

A practical path planning methodology for wire and arc additive manufacturing of thin-walled structures

Voronoi diagram computations for planar NURBS curves

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