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Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee

Abstract

The medial axis of a surface in 3D is the closure of all points that have two or more closest points on the surface. It is an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue to improve their approximations. Voronoi diagrams turn out to be useful for this approximation. Although it is known that Voronoi vertices for a sample of points from a curve in 2D approximate its medial axis, a similar result does not hold in 3D. Recently, it has been discovered that only a subset of Voronoi vertices converge to the medial axis as sample density approaches infinity. However, most applications need a nondiscrete approximation as opposed to a discrete one. To date no known algorithm can compute this approximation straight from the Voronoi diagram with a guarantee of convergence. We present such an algorithm and its convergence analysis in this paper. One salient feature of the algorithm is that it is scale and density independent. Experimental results corroborate our theoretical claims.

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Correspondence to Tamal K. Dey or Wulue Zhao.

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Dey, T., Zhao, W. Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee. Algorithmica 38, 179–200 (2004). https://doi.org/10.1007/s00453-003-1049-y

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  • Medial axis
  • Geometric modeling
  • Samples
  • Voronoi diagram
  • Delaunay triangulation