Voronoi cells in metric algebraic geometry of plane curves (Q6561305)

Metric algebraic geometry addresses questions about real algebraic varieties involving distances. In the paper, the authors use Voronoi cells and their duals, Delaunay cells, to study metric features of plane curves.\N\NGiven a point \(x\) on a real algebraic plane curve \(X \subseteq \mathbb{R}^2\), the \textit{Voronoi cell} of \(X\) at \(x\) is the locus of points that are closer to \(x\) than to any other point of \(X\). The \textit{medial axis} of a variety is the locus of points that have more than one nearest point on \(X\). Given a point in \(\mathbb{R}^2\), one may ask how close it must be to \(X\) to have a unique nearest point on \(X\). The infimum of the set of distances from any point on \(X\) to a point on the medial axis of \(X\) is called the reach of \(X\).\N\NIn the paper, the authors 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. This result is then used to study various metric features of plane curves, including the medial axis, curvature, evolute, bottlenecks, and reach. They show how to identify each desired metric feature from Voronoi or Delaunay cells, enabling approximation of these features by finite point samples from the variety. The paper also includes an algorithm that computes Voronoi and Delaunay cells of sampled points to estimate the reach of the curve.