We present medial parametrization, a new approach to parameterizing any compact planar domain bounded by simple closed curves. The basic premise behind our proposed approach is to use two close Voronoi sites, which we call dipoles, to construct and reconstruct an approximate piecewise-linear version of the original boundary and medial axis through Voronoi tessellation. The boundaries and medial axes of such planar compact domains offer a natural way to describe the domain's interior. Any compact planar domain is homeomorphic to a compact unit circular disk admits a natural parameterization isomorphic to the polar parametrization of the disk. Specifically, the medial axis and the boundary generalize the radial and angular parameters, respectively. In this paper, we present a simple algorithm that puts these principles into practice. The algorithm is based on the simultaneous re-creation of the boundaries of the domain and its medial axis using Voronoi tessellation. This simultaneous re-creation provides partitions of the domain into a set of "skinny" convex polygons wherein each polygon is essentially a subset of the medial edges (which we call the spine) connected to the boundary through exactly two straight edges (which we call limbs). This unique structure enables us to convert the original Voronoi tessellation into quadrilaterals and triangles (at the poles of the medial axis) neatly ordered along the domain boundary, thereby allowing proper parametrization of the domain. Our approach is agnostic to the number of holes and disconnected components bounding the domain. We investigate the efficacy of our concept and algorithm through several examples.

翻译：我们提出中轴参数化——一种对任意由简单闭曲线围成的平面紧致域进行参数化的新方法。该方法的核心思想是利用两个邻近的Voronoi站点（称为偶极子），通过Voronoi细分来构造与重构原始边界及中轴的近似分段线性版本。此类平面紧致域的边界与中轴能为域内部提供自然的描述方式。任意平面紧致域均与单位圆盘同胚，因此存在一种自然参数化，其形式与圆盘的极坐标参数化同构。具体而言，中轴与边界分别推广了径向与角度参数。本文提出一个实现上述原理的简洁算法，该算法基于利用Voronoi细分同时重构域边界及其内部中轴的过程。这种同步重构将域划分为一组"细长"凸多边形——每个多边形本质上是中轴边（称为脊柱）的子集，并通过两条直边（称为肢干）与边界相连。这种独特结构使我们能够将原始Voronoi细分转换为沿域边界有序排列的四边形与三角形（位于中轴极点处），进而实现对域的恰当参数化。本方法对围成域的孔洞数量及不连通分量具有无关性。我们通过多个实例验证了所提概念与算法的有效性。