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MiS Preprint
44/2019

Voronoi Cells in Metric Algebraic Geometry of Plane Curves

Madeline Brandt and Madeleine Weinstein

Abstract

Voronoi cells of varieties encode many features of their metric geometry. We prove that each Voronoi or Delaunay cell of a plane curve appears as the limit of a sequence of cells obtained from point samples of the curve. We then use this result to study metric features of plane curves, including the medial axis, curvature, evolute, bottlenecks, and reach. In each case, we provide algebraic equations defining the object and, where possible, give formulas for the degrees of these algebraic varieties. We then show how to identify the desired metric feature from Voronoi or Delaunay cells, and therefore how to approximate it by a finite point sample from the variety.

Received:
01.07.2019
Published:
10.07.2019

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Preprint
2019 Repository Open Access
Madeline Brandt and Madeleine Weinstein

Voronoi cells in metric algebraic geometry of plane curves