Computation of Voronoi Diagrams of Circular Arcs and Straight Lines

Vroni is one of few existing implementations for the stable computation of Voronoi diagrams of line segments. A topology-oriented approach in combination with double-precision floating-point arithmetic makes Vroni also the fastest and most reliable implementation available. Up to now, Voronoi diagram algorithms used in industrial applications process input data consisting of points and straightline segments. Since circular arcs are important in various applications like CAD/CAM, printed circuit boards, etc., so far circular arcs have been approximated by a reasonable number of straight-line segments. Extending the algorithm to support circular arcs has several advantages over an approximated solution, including higher performance and lower memory consumption. In this diploma thesis we extend Vroni to genuine circular arcs, pursuing the strategy of topological constraints and carefully implemented numerical procedures based on double-precision floating-point arithmetic. To the best of our knowledge, this makes Vroni the first implementation supporting genuine circular arcs. We provide an extensive mathematical analysis including proofs of correctness, computation of Voronoi nodes and compare the new implementation with the pre-genuine-arc version of Vroni.

[1]  M. Karavelas.  A robust and efficient implementation for the segment Voronoi diagram , 2004 .

[2]  R. Seidel.  Backwards Analysis of Randomized Geometric Algorithms , 1993 .

[3]  D. T. Lee,et al.  Medial Axis Transformation of a Planar Shape , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[4]  Otfried Cheong,et al.  The Voronoi Diagram of Curved Objects , 1995, SCG '95.

[5]  Martin Held,et al.  Biarc Approximation, Simplification and Smoothing of Polygonal Curves by Means of Voronoi-Based Tolerance Bands , 2008, Int. J. Comput. Geom. Appl..

[6]  Deok-Soo Kim,et al.  Voronoi diagram of a circle set from Voronoi diagram of a point set: I. Topology , 2001, Computer Aided Geometric Design.

[7]  Toshiyuki Imai,et al.  A Topology Oriented Algorithm for the Voronoi Diagram of Polygons , 1996, Canadian Conference on Computational Geometry.

[8]  Chee-Keng Yap,et al.  AnO(n logn) algorithm for the voronoi diagram of a set of simple curve segments , 1987, Discret. Comput. Geom..

[9]  Deok-Soo Kim,et al.  A sweepline algorithm for Euclidean Voronoi diagram of circles , 2006, Comput. Aided Des..

[10]  M. Iri,et al.  Construction of the Voronoi diagram for 'one million' generators in single-precision arithmetic , 1992, Proc. IEEE.

[11]  D. T. Lee,et al.  Generalization of Voronoi Diagrams in the Plane , 1981, SIAM J. Comput..

[12]  Gábor Lukács,et al.  Pocket machining based on contour-parallel tool paths generated by means of proximity maps , 1994, Comput. Aided Des..

[13]  Kokichi Sugihara,et al.  Topology-Oriented Implementation—An Approach to Robust Geometric Algorithms , 2000, Algorithmica.

[14]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[15]  Martin Held,et al.  VRONI: An engineering approach to the reliable and efficient computation of Voronoi diagrams of points and line segments , 2001, Comput. Geom..

[16]  Steven Fortune,et al.  A sweepline algorithm for Voronoi diagrams , 1986, SCG '86.

[17]  Micha Sharir,et al.  Intersection and Closest-Pair Problems for a Set of Planar Discs , 1985, SIAM J. Comput..

[18]  Martin Held,et al.  On the Computational Geometry of Pocket Machining , 1991, Lecture Notes in Computer Science.

[19]  Kurt Mehlhorn,et al.  Randomized Incremental Construction of Abstract Voronoi Diagrams , 1993, Comput. Geom..

[20]  Joseph S. B. Mitchell,et al.  PVD: A Stable Implementation for Computing Voronoi Diagrams of Polygonal Pockets , 2001, ALENEX.

[21]  Mariette Yvinec,et al.  The Voronoi Diagram of Planar Convex Objects , 2003, ESA.

[22]  Mariette Yvinec,et al.  Dynamic Additively Weighted Voronoi Diagrams in 2D , 2002, ESA.

[23]  Martin Held,et al.  Heuristics for the Generation of Random Polygons , 1996, CCCG.

[24]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[25]  L. Kucera,et al.  Randomized incremental construction of abstract Voronoi diagrams , 1993 .

[26]  Rolf Klein,et al.  Concrete and Abstract Voronoi Diagrams , 1990, Lecture Notes in Computer Science.