The union of balls and its dual shape

Efficient algorithms are described for computing topological, combinatorial, and metric properties of the union of finitely many spherical balls in ℝd. These algorithms are based on a simplicial complex dual to a decomposition of the union of balls using Voronoi cells, and on short inclusion-exclusion formulas derived from this complex. The algorithms are most relevant in ℝ3 where unions of finitely many balls are commonly used as models of molecules.

[1]  G. L. Dirichlet.  Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. , 1850 .

[2]  Georges Voronoi.  Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire. Sur quelques propriétés des formes quadratiques positives parfaites. , 1908 .

[3]  J. Rotman.  An Introduction to Algebraic Topology , 1957 .

[4]  M. Fuchs.  A note on mapping cylinders. , 1971 .

[5]  F M Richards,et al.  Areas, volumes, packing and protein structure. , 1977, Annual review of biophysics and bioengineering.

[6]  Ravi Kannan,et al.  Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix , 1979, SIAM J. Comput..

[7]  David G. Kirkpatrick,et al.  On the shape of a set of points in the plane , 1983, IEEE Trans. Inf. Theory.

[8]  M. L. Connolly.  Analytical molecular surface calculation , 1983 .

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

[10]  Franz Aurenhammer,et al.  Improved Algorithms for Discs and Balls Using Power Diagrams , 1988, J. Algorithms.

[11]  Paul D. Domich.  Residual hermite normal form computations , 1989, TOMS.

[12]  Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms , 1990, TOGS.

[13]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[14]  Herbert Edelsbrunner,et al.  Weighted alpha shapes , 1992 .

[15]  Incremental topological flipping works for regular triangulations , 1992, SCG '92.

[16]  D. Naiman,et al.  INCLUSION-EXCLUSION-BONFERRONI IDENTITIES AND INEQUALITIES FOR DISCRETE TUBE-LIKE PROBLEMS VIA EULER CHARACTERISTICS , 1992 .

[17]  Herbert Edelsbrunner,et al.  An incremental algorithm for Betti numbers of simplicial complexes , 1993, SCG '93.

[18]  Herbert Edelsbrunner,et al.  Three-dimensional alpha shapes , 1994, ACM Trans. Graph..

[19]  Herbert Edelsbrunner,et al.  Measuring proteins and voids in proteins , 1995, Proceedings of the Twenty-Eighth Annual Hawaii International Conference on System Sciences.