Given two simplicial complexes \({\mathcal C}_{\rm 1}\) and \({\mathcal C}_{\rm 2}\) embedded in Euclidean space \({\mathbb R}^{d}\), \({\mathcal C}_{\rm 1}\) subdivides \({\mathcal C}_{\rm 2}\) if (i) \({\mathcal C}_{\rm 1}\) and \({\mathcal C}_{\rm 2}\) have the same underlying space, and (ii) every simplex in \({\mathcal C}_{\rm 1}\) is contained in a simplex in \({\mathcal C}_{\rm 2}\). In this paper we present a method to compute a set of weighted points whose alpha complex subdivides a given simplicial complex.

Following this, we also show a simple method to approximate a given polygonal object with a set of balls via computing the subdividing alpha complex of the boundary of the object. The approximation is robust and is able to achieve a union of balls whose Hausdorff distance to the object is less than a given positive real number ε.


Convex Hull Simplicial Complex Voronoi Cell Weighted Point Underlying Space 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Ho-lun Cheng
    • 1
  • Tony Tan
    • 1
  1. 1.School of ComputingNational University of Singapore 

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