Journal of KIISE:Computer Systems and Theory (한국정보과학회논문지:시스템및이론)

Korean Institute of Information Scientists and Engineers (한국정보과학회)
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An Implementation of an Edge-based Algorithm for Separating and Intersecting Spherical Polygons

구 볼록 다각형 들의 분리 및 교차를 위한 간선 기반 알고리즘의 구현

  • Ha, Jong-Seong (Dept.of Information Communication Computer Engineering, Woosuk University) ;
  • Cheon, Eun-Hong (Dept.of Information Communication Computer Engineering, Woosuk University)
  • 하종성 (우석대학교 정보통신컴퓨터공학부) ;
  • 천은홍 (우석대학교 정보통신컴퓨터공학부)
  • Published : 2001.10.01
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Abstract

In this paper, we consider the method of partitioning a sphere into faces with a set of spherical convex polygons $\Gamma$=$｛P_1...P_n｝$ for determining the maximum of minimum intersection. This problem is commonly related with five geometric problems that fin the densest hemisphere containing the maximum subset of $\Gamma$, a great circle separating $\Gamma$, a great circle bisecting $\Gamma$ and a great circle intersecting the minimum or maximum subset of $\Gamma$. In order to efficiently compute the minimum or maximum intersection of spherical polygons. we take the approach of edge-based partition, in which the ownerships of edges rather than faces are manipulated as the sphere is incrementally partitioned by each of the polygons. Finally, by gathering the unordered split edges with the maximum number of ownerships. we approximately obtain the centroids of the solution faces without constructing their boundaries. Our algorithm for finding the maximum intersection is analyzed to have an efficient time complexity O(nv) where n and v respectively, are the numbers of polygons and all vertices. Furthermore, it is practical from the view of implementation, since it computes numerical values. robustly and deals with all the degenerate cases, Using the similar approach, the boundary of a general intersection can be constructed in O(nv+LlogL) time, where : is the output-senstive number of solution edges.

본 논문에서는 구상에서 주어진 볼록 다각형의 집합$\Gamma$=$｛P_1...P_n｝$의 최대 또는 최소 교차를 결정하기 위하여 다각형의 간선으로 구를 면으로 분할하는 문제를 고려한다. 이 문제는 $\Gamma$의 최대 부분집합을 포함하는 반구를 $\Gamma$를 분리하는 대원을, $\Gamma$를 이분하는 대원을 $\Gamma$를 최소 또는 최대 부분집합을 교차하는 대원을 각각 찾는 다섯가지 기하적 문제를 공통적으로 관련이 있다. 구다각형의 최대 및 최소 교차를 효율적으로 구하기 위하여 우리는 간선 기반 분할의 방식을 취하는데 이 방식에서는 구가 각 다각형에 의해 증분적으로 분할되면서 면이 아닌 면을 구성하는 간선의 소유권이 처리된다. 마지막에는 최대수의소유권을 가지는 분할된 비정렬 간선들을 모아 해가 되는 면들의 경계를 구성하지 않고 그들의 중심을 근사적으로 얻는다. 최대 교차를 찾는 우리의 알고리즘은 효율적인 시간복잡도 O(nv)를 가지는 것으로 분석된다. 여기서 n는 v은 각각 다각형과 모든 장점의 개수들이다. 더구나 견고하게 수치를 계산하고 모든 degeneracy 경우를 다루기 때문에 구현의 관점에서도 실제적이다. 유사한 방식을 사용하여 일반적인 교차의 모든 경계는 O(nv+LlogL)시간에 구성할 수 있다. 여기서 L은 해로 출력되는 간선의 개수이다.

Keywords

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