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Voronoi cells of varieties encode many features of their metric geometry. Weprove that each Voronoi or Delaunay cell of a plane curve appears as the limitof a sequence of cells obtained from point samples of the curve. We use thisresult to study metric features of plane curves, including the medial axis,curvature, evolute, bottlenecks, and reach. In each case, we provide algebraicequations defining the object and, where possible, give formulas for thedegrees of these algebraic varieties. We show how to identify the desiredmetric feature from Voronoi or Delaunay cells, and therefore how to approximateit by a finite point sample from the variety.

Voronoi Cells in Metric Algebraic Geometry of Plane Curves