An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere

Abstract A general and direct method for computing the Betti numbers of a finite simplicial complex in S d is given. This method is complete for d ⩽ 3, where versions of this method run in time O(nα(n)) and O(n), n the number of simplices. An implementation of the algorithm is applied to alpha shapes, which is a novel geometric modeling tool.

[1]  Henry J. Stephen Smith,et al.  XV. On systems of linear indeterminate equations and congruences , 1861, Philosophical Transactions of the Royal Society of London.

[2]  Marshall W. Bern,et al.  Compatible tetrahedralizations , 1993, SCG '93.

[3]  Martti Mäntylä,et al.  Introduction to Solid Modeling , 1988 .

[4]  P. Giblin.  Graphs, surfaces, and homology , 1977 .

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

[6]  Christoph M. Hoffmann,et al.  Geometric and Solid Modeling , 1989 .

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

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

[9]  Herbert Edelsbrunner,et al.  Three-dimensional alpha shapes , 1992, VVS.

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

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

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

[13]  Christoph M. Hoffmann,et al.  Geometric and Solid Modeling: An Introduction , 1989 .

[14]  B.R. Donald,et al.  On the complexity of computing the homology type of a triangulation , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

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