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：本文提出中轴参数化（medial parametrization），一种对由简单闭合曲线界定的任意紧致平面区域进行参数化的新方法。该方法的核心思想是使用两个紧密排列的Voronoi站点（我们称之为偶极子），通过Voronoi剖分来构建并重构原始边界及中轴的近似分段线性版本。此类平面紧致区域的边界与中轴为描述区域内部结构提供了自然途径。任意紧致平面区域与紧致单位圆盘同胚，可建立与圆盘极坐标参数化同构的自然参数化——具体而言，中轴与边界分别推广了径向参数与角度参数。本文提出一种实现该原理的简洁算法：利用Voronoi剖分同时重构区域边界及其内部中轴。这种同步重构将区域分割为一组"瘦长"凸多边形，其中每个多边形实质上是中轴边（称为脊）通过恰好两条直边（称为肢）连接至边界的子集。该独特结构使原始Voronoi剖分能转化为沿区域边界有序排列的四边形与（中轴极点处的）三角形单元，从而实现区域的有效参数化。本方法对区域边界所包含的孔洞数量及非连通分量均保持无关性。通过多个算例验证了所提概念与算法的有效性。