# The union of balls and its dual shape

@article{Edelsbrunner1993TheUO, title={The union of balls and its dual shape}, author={Herbert Edelsbrunner}, journal={Discrete \& Computational Geometry}, year={1993}, volume={13}, pages={415-440} }

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.

## 425 Citations

### Computing polygonal surfaces from unions of balls

- Computer ScienceProceedings Computer Graphics International, 2004.
- 2004

A new algorithm is presented for computing a polygonal surface from a union of balls that uses the dual shape of the balls to give the resulting surface the correct topology.

### Contemporary Mathematics State of the Union ( of Geometric Objects )

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Given a sample of points from the boundary of an object in IR 3, a representation of the object as a union of balls is constructed, and it is shown that the set of ball centers in the construction converges to the true medial axis as the sampling density increases.

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An output-sensitive algorithm to compute α-complexes of n-point sets in constant dimensions, whose running time is O(f log n log αs), where s is the smallest pairwise distance and f is the number of simplices in the cα-complex for a constant c.

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