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Medial Parametrization of Arbitrary Planar Compact Domains with Dipoles

Vinayak Krishnamurthy, Ergun Akleman

TL;DR

This work introduces medial parametrization, a framework to parameterize arbitrary compact planar domains bounded by simple closed curves by using two nearby dipoles to jointly reconstruct the boundary and medial axis via Voronoi tessellation. The method converts the resulting structure into a quad-dominant remeshing, enabling a straightforward boundary-to-medial-axis parameterization that generalizes polar coordinates beyond simply-connected disks. It is robust to multiple components and holes, demonstrated through multiple examples, and offers a practical path from geometric primitives (boundary and skeleton) to a usable coordinate system for sampling and meshing. The approach has potential applications in GIS, architectural geometry, and texture mapping, and points toward extensions to open curves, curve networks, wallpaper domains, and even higher dimensions.

Abstract

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.

Medial Parametrization of Arbitrary Planar Compact Domains with Dipoles

TL;DR

This work introduces medial parametrization, a framework to parameterize arbitrary compact planar domains bounded by simple closed curves by using two nearby dipoles to jointly reconstruct the boundary and medial axis via Voronoi tessellation. The method converts the resulting structure into a quad-dominant remeshing, enabling a straightforward boundary-to-medial-axis parameterization that generalizes polar coordinates beyond simply-connected disks. It is robust to multiple components and holes, demonstrated through multiple examples, and offers a practical path from geometric primitives (boundary and skeleton) to a usable coordinate system for sampling and meshing. The approach has potential applications in GIS, architectural geometry, and texture mapping, and points toward extensions to open curves, curve networks, wallpaper domains, and even higher dimensions.

Abstract

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.
Paper Structure (29 sections, 4 equations, 16 figures)

This paper contains 29 sections, 4 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Context & Motivation
  3. Basis & Rationale
  4. Approach
  5. Contributions
  6. Background
  7. Barycentric Coordinates
  8. Medial Axis Computation
  9. Boundary-Skeleton Co-computation
  10. Meshing
  11. Our work
  12. Conceptual Preliminaries
  13. Polar Parametrization
  14. Medial Parametrization
  15. Computing Medial Parametrization
  16. ...and 14 more sections

Figures (16)

  • Figure 1: An example that demonstrates the whole process for a domain that consists of three disconnected regions.
  • Figure 2: An example that demonstrates the parameterization of a compact planar domain bounded by a simple (non-self-intersecting) curve. Note that our re-meshing stage is a subdivision algorithm that smooths both the boundary curve and medial axis.
  • Figure 3: An example that demonstrates the whole process for a domain with three holes.
  • Figure 4: A conceptual example of medial parametrization
  • Figure 5: Nomenclature for the medial parametrization algorithm.
  • ...and 11 more figures