Medial Parametrization of Arbitrary Planar Compact Domains with Dipoles
CoRR(2024)
摘要
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.
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