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 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 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.

翻译：簇的Voronoi细胞编码了其度量几何的许多特征。我们证明，平面曲线的每个Voronoi或Delaunay细胞都表现为从曲线点样本获得的细胞序列的极限。我们利用这一结果研究平面曲线的度量特征，包括中轴、曲率、渐屈线、瓶颈及可达距离。在每种情况下，我们提供定义这些对象的代数方程，并在可能时给出这些代数簇的次数公式。我们展示如何从Voronoi或Delaunay细胞中识别所需度量特征，从而通过簇的有限点样本来逼近该特征。