Limits of Voronoi Diagrams

In this thesis we study sets of points in the plane and their Voronoi diagrams, in particular when the points coincide. We bring together two ways of studying point sets that have received a lot of attention in recent years: Voronoi diagrams and compactifications of configuration spaces. We study moving and colliding points and this enables us to introduce `limit Voronoi diagrams'. We define several compactifications by considering geometric properties of pairs and triples of points. In this way we are able to define a smooth, real version of the Fulton-MacPherson compactification. We show how to define Voronoi diagrams on elements of these compactifications and describe the connection with the limit Voronoi diagrams.

[1]  James W. Walker,et al.  Homotopy Type and Euler Characteristic of Partially Ordered Sets , 1981, Eur. J. Comb..

[2]  W. Fulton,et al.  A compactification of configuration spaces , 1994 .

[3]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[4]  Joachim Gudmundsson,et al.  Higher order Delaunay triangulations , 2000, Comput. Geom..

[5]  Franz Aurenhammer,et al.  A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams , 1992, Int. J. Comput. Geom. Appl..

[6]  Leonidas J. Guibas,et al.  Voronoi Diagrams of Moving Points , 1998, Int. J. Comput. Geom. Appl..

[7]  Robert Ghrist,et al.  Finding Topology in a Factory: Configuration Spaces , 2002, Am. Math. Mon..

[8]  M. Berger,et al.  Differential Geometry: Manifolds, Curves, and Surfaces , 1987 .

[9]  Chee-Keng Yap,et al.  Robust Geometric Computation , 2016, Encyclopedia of Algorithms.

[10]  R. Pollack,et al.  Advances in Discrete and Computational Geometry , 1999 .

[11]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[12]  K. D. Joshi.  Introduction to General Topology , 1983 .

[13]  Deformations of algebras over operads and Deligne's conjecture , 2000, math/0001151.

[14]  J. van Leeuwen,et al.  Discrete and Computational Geometry , 2002, Lecture Notes in Computer Science.

[15]  D. Kendall.  A Survey of the Statistical Theory of Shape , 1989 .

[16]  Boris Aronov,et al.  Results on k-sets and j-facets via continuous motion , 1998, SCG '98.

[17]  Franz Aurenhammer,et al.  A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams , 1991, SCG '91.

[18]  D. T. Lee,et al.  On k-Nearest Neighbor Voronoi Diagrams in the Plane , 1982, IEEE Transactions on Computers.

[19]  A. Björner.  Topological methods , 1996 .

[20]  Satyan L. Devadoss.  A Space of Cyclohedra , 2001, Discret. Comput. Geom..

[21]  M. Kontsevich.  Operads and Motives in Deformation Quantization , 1999, math/9904055.

[22]  G. Ziegler.  Lectures on Polytopes , 1994 .

[23]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[24]  Georges Voronoi.  Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs. , 1908 .

[25]  H. C. Corben,et al.  Classical Mechanics (2nd ed.) , 1961 .

[26]  Micha Sharir.  k-sets and random hulls , 1993, Comb..

[27]  John H. Conway,et al.  The sensual (quadratic) form , 1997 .

[28]  Satyan L. Devadoss.  Tessellations of Moduli Spaces and the Mosaic Operad , 1998, math/9807010.

[29]  Herbert Edelsbrunner,et al.  Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms , 1988, SCG '88.

[30]  Herbert Edelsbrunner,et al.  Geometry and Topology for Mesh Generation , 2001, Cambridge monographs on applied and computational mathematics.

[31]  H. Coxeter,et al.  Generators and relations for discrete groups , 1957 .

[32]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[33]  P. Orlik,et al.  Arrangements Of Hyperplanes , 1992 .

[34]  E. Rees.  Notes on Geometry , 1983 .