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Representing the Voronoi diagram of a simple polygon using rational quadratic Bézier curves

Elsevier Ltd
Publication Date
DOI: 10.1016/0010-4485(95)99797-c
  • Voronoi Diagrams
  • Curves
  • Rational Polynomials
  • Computer Science
  • Mathematics


Abstract The Voronoi diagram of a set of geometric entities on a plane, such as points, line segments, or arcs, is a collection of Voronoi polygons associated with each entity, where the Voronoi polygon of an entity is a set of points which are closer to the associated entity than any other entity. A Voronoi diagram is one of the most fundamental geometrical constructs, and it is well known for its theoretical elegance and the wealth of applications. Various geometric problems can be solved with the aid of Voronoi diagrams. The paper discusses an algorithm to construct the Voronoi diagram of the interior of a simple polygon which consists of simple curves such as line segments as well as arcs in a plane with O( N log N) time complexity by the use of a divide-and-conquer scheme. Particular emphasis is placed on the parameterization of bisectors using a rational quadratic Bézier curve representation which unifies four different bisector cases.

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