Three-dimensional alpha shapes
ACM Transactions on Graphics (TOG)
Algorithmic geometry
On the definition and the construction of pockets in macromolecules
Discrete Applied Mathematics - Special volume on computational molecular biology DAM-CMB series volume 2
Approximate medial axis as a voronoi subcomplex
Proceedings of the seventh ACM symposium on Solid modeling and applications
Complexity of the delaunay triangulation of points on surfaces the smooth case
Proceedings of the nineteenth annual symposium on Computational geometry
Any open bounded subset of Rn has the same homotopy type than its medial axis
SM '03 Proceedings of the eighth ACM symposium on Solid modeling and applications
Provably good surface sampling and approximation
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
A Linear Bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces
Discrete & Computational Geometry
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Graphical Models
Geometry and Topology for Mesh Generation (Cambridge Monographs on Applied and Computational Mathematics)
Medial axis approximation and unstable flow complex
Proceedings of the twenty-second annual symposium on Computational geometry
Stability and homotopy of a subset of the medial axis
SM '04 Proceedings of the ninth ACM symposium on Solid modeling and applications
The theory of multidimensional persistence
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Convex hull and voronoi diagram of additively weighted points
ESA'05 Proceedings of the 13th annual European conference on Algorithms
The power crust, unions of balls, and the medial axis transform
Computational Geometry: Theory and Applications
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Computation of more channels in protein molecules
EG VCBM'08 Proceedings of the First Eurographics conference on Visual Computing for Biomedicine
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Given a collection β of balls in three-dimensional space, each having a radius of at least 1, we present an approximation scheme that constructs a collection Kε of unit balls that approximate β, such that the Hausdorff distance between ∪β and ∪Kε is at most ε. We define the pathway axis as the subset of the medial axis of the complement of ∪β for which the set of closest balls in β do not have a common intersection. It is the medial axis of the complement of ∪β without `dead-ends' and therefore it is a good starting point for finding pathways that lie outside ∪β. The recently introduced persistence diagram of the distance function from ∪β encodes topological characteristics of the function, giving a measure on the importance of topological features such as voids or tunnels during a uniform growth process of β. In this paper we introduce the pathway diagram as a useful subset of the Voronoi diagram of the centers of the unit balls in Kε, which can be easily and efficiently computed. We show that the pathway diagram contains an approximation of the pathway axis of β. We prove a bound on the ratio |Kε|/|β|, namely the ratio between the number of unit balls in Kε and the number of balls in β. We employ this bound to show how we efficiently approximate the persistence diagram of ∪β. Finally, we show that our approach is superior to the standard point-sample approaches for the two problems that we address in this paper: Approximating the medial axis of the complement of ∪β, and approximating the persistence diagram of ∪β. In a companion paper we introduce MolAxis, a tool for the identification of channels in macromolecules, that demonstrates how the pathway diagram and the persistence diagram are used to identify pathways in the complement of molecules.