Korean Journal of Computational Design and Engineering (한국CDE학회논문집)

Society for Computational Design and Engineering (한국CDE학회)
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The Computation of the Voronoi Diagram of a Circle Set Using the Voronoi Diagram of a Point Set: II. Geometry

점 집합의 보로노이 다이어그램을 이용한 원 집합의 보로노이 다이어그램의 계산: II.기하학적 측면

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  • Kokichi Sugihara (University of Tokyo, Department of Mathematical Engineering and Information Physics)
  • Published : 2001.03.01
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Abstract

Presented in this paper are algorithms to compute the positions of vertices and equations of edges of the Voronoi diagram of a circle set. The circles are located in a Euclidean plane, the radii of the circles are not necessarily equal and the circles are not necessarily disjoint. The algorithms correctly and efficiently work when the correct topology of the Voronoi diagram was given. Given three circle generators, the position of the Voronoi vertex is computed by treating the plane as a complex plane, the Z-plane, and transforming it into another complex plane, the W-plane, via the Mobius transformation. Then, the problem is formulated as a simple point location problem in regions defined by two lines and two circles in the W-plane. And the center of the inverse-transformed circle in Z-plane from the line in the W-plane becomes the position of the Voronoi vertex. After the correct topology is constructed with the geometry of the vertices, the equations of edge are computed in a rational quadratic Bezier curve farm.

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