53237055
Three-dimensional alpha shapes
Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the “shape” of the set. For that purpose, this article introduces the formal notion of the family of α-shapes of a finite point set in R 3 . Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter α e R controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size n in time 0(n 2 ) , worst case. A robust implementation of the algorithm is discussed, and several applications in the area of scientific computing are mentioned.
9599628
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 ballsin <inline-equation><f><blkbd>R</blkbd><sup>d</sup></f></inline-equation>. These algorithms are based on a simplicial complexdual to a certain decomposition of the union of balls, and on shortinclusion-exclusion formulas derived from this complex. The algorithmsare most relevant in <inline-equation><f><blkbd>R</blkbd><sup>3</sup></f></inline-equation> <?Pub Caret>where unions of finitely many balls arecommonly used as models of molecules.
6983029
On the shape of a set of points in the plane
A generalization of the convex hull of a finite set of points in the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, " \alpha -shapes," which seem to capture the intuitive notions of "fine shape" and "crude shape" of point sets. It is shown that a-shapes are subgraphs of the closest point or furthest point Delaunay triangulation. Relying on this result an optimal O(n \log n) algorithm that constructs \alpha -shapes is developed.
444430
Loops in reeb graphs of 2-manifolds
Given a Morse function f over a 2-manifold with or without boundary, the Reeb graph is obtained by contracting the connected components of the level sets to points. We prove tight upper and lower bounds on the number of loops in the Reeb graph that depend on the genus, the number of boundary components, and whether or not the 2-manifold is orientable. We also give an algorithm that constructs the Reeb graph in time O(nlogn), where n is the number of edges in the triangulation used to represent the 2-manifold and the Morse function.
3265202
Emerging Challenges in Computational Topology
Here we present the results of the NSF-funded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology.
46103497
Loops in Reeb Graphs of 
2-Manifolds
Abstract
 Given a Morse function f over a 2-manifold with or without boundary,
 the Reeb graph is obtained by contracting the connected components
 of the level sets to points.
 We prove tight upper and lower bounds on the
 number of loops in the Reeb graph
 that depend on the genus, the number of boundary components,
 and whether or not the 2-manifold is orientable.
 We also give an algorithm that constructs the Reeb graph in time
 O(n log n), where n is the number of edges in the
 triangulation used to represent the 2-manifold and the Morse function.


14707519
Inequalities for the curvature of curves and surfaces
In this paper, we bound the difference between the total mean curvatures of two closed surfaces in R3 in terms of their total absolute curvatures and the Fréchet distance between the volumes they enclose. The proof relies on a combination of methods from algebraic topology and integral geometry. We also bound the difference between the lengths of two curves using the same methods.
30416185
On the Zone Theorem for Hyperplane Arrangements
The zone theorem for an arrangement of n hyperplanes in d-dimensional real space says that the total number of faces bounding the cells intersected by another hyperplane is $O(n^{d - 1} )$. This result is the basis of a time-optimal incremental algorithm that constructs a hyperplane arrangement and has a host of other algorithmic and combinatorial applications. Unfortunately, the original proof of the zone theorem, for $d \geqslant 3$, turned out to contain a serious and irreparable error. This paper presents a new proof of the theorem. The proof is based on an inductive argument, which also applies in the case of pseudohyperplane arrangements. The fallacies of the old proof along with some ways of partially saving that approach are briefly discussed.