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

한국정보과학회 (Korean Institute of Information Scientists and Engineers)
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구 볼록 다각형 들의 분리 및 교차를 위한 간선 기반 알고리즘의 구현

An Implementation of an Edge-based Algorithm for Separating and Intersecting Spherical Polygons

  • 하종성 (우석대학교 정보통신컴퓨터공학부) ;
  • 천은홍 (우석대학교 정보통신컴퓨터공학부)
  • Ha, Jong-Seong (Dept.of Information Communication Computer Engineering, Woosuk University) ;
  • Cheon, Eun-Hong (Dept.of Information Communication Computer Engineering, Woosuk University)
  • 발행 : 2001.10.01
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⟨ 이전 논문 다음 논문 ⟩

초록

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.

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.

키워드

참고문헌

  1. L. L. Chen, S. Y. Chou, and T. C. Woo, 'Separating and intersecting spherical polygons: computing machinability on three- four- and five-axis numerically controlled machines,' ACM Tr. on Graphics, Vol. 12, No. 14, pp.305-326, 1993 https://doi.org/10.1145/159730.159732
  2. S. H. Suh and J. K. Kang, 'Process planning for multi-axis NC machining of free surfaces,' Int. J. Prod. Res. Vol. 33, No. 10, pp.2723-2738, 1995 https://doi.org/10.1080/00207549508904841
  3. J. W. H. Tangelder, 'Automated fabrication of shape models of free-form objects with a sculpturing robot,' Ph. D. Thesis, Industrial Design Engineering of Delft Univ. of Technology, 1998
  4. P. Gupta et al., 'Efficient geometric algorithms for workpiece orientation in 4- and 5-axis NC machining,' Computer-Aided Design, Vol. 28, No. 8, pp.577-587, 1996 https://doi.org/10.1016/0010-4485(95)00071-2
  5. K. Tang, T. Woo, and J. Gan, 'Maximum intersection of spherical polygons and workpiece orientation for 4- and 5-axis machining,' ASME J. Mechanical Design, Vol. 114, pp. 477-485, 1992
  6. B. G. Baumgart, 'A polyhedron representation for computer vision,' In National Computer Conference, Anaheim, CA, IFIPS, pp.589-596, 1975
  7. K. Weiler, 'Edge-based data structures for solid modeling in curved-surface environments,' IEEE Computer Graphics and Application, Vol. 5, No.1, pp.21-40, 1985
  8. T. C. Woo, 'A combinatorial analysis of boundary data structure schema,' IEEE Comput. Graph Appl., Vol. 5, No.3, pp.19-27, 1985
  9. S. M. Barker, 'Towards a topology for computational geometry,' Computer-Aided Design, Vol. 27, No.4, pp.311-318, 1995 https://doi.org/10.1016/0010-4485(95)91141-7
  10. S. Fortune, 'Efficient exact arithmetic for computational geometry,' Proc. 9th ACM Symp. Comp. Geom., pp.163-172, 1993 https://doi.org/10.1145/160985.161015
  11. C. M. Hoffmann, 'The problems of accuracy and robustness in geometric computation,' Computer, Vol. 22, No.3, pp.31-41, 1989 https://doi.org/10.1109/2.16223
  12. V. Milenkovic, 'Verifiable implementations of geometric algorithms using finite precision arithmetic,' Artificial Intelligence, Vol. 37, pp.377-401, 1988 https://doi.org/10.1016/0004-3702(88)90061-6
  13. E. Horowitz and S. Sahni, Fundamentals of Data Structure, Computer Science Press, 1973
  14. J. G. Gan, T. C. Woo and K. Tang, 'Spherical maps: their construction, properties, and approximation,' ASME J. Mechanical Design, Vol. 116, pp.357-363, 1994
  15. L. L. Chen and T. C. Woo, 'Computational geometry on the sphere with application to automated machining,' Tr. ASME, Vol. 114, pp. 288-295, 1992
  16. M. E. Dyer, 'Linear time algorithms for two- and three-variable linear programs,' SIAM J. Computing, Vol. 13, No. 1, pp.31-45, 1984 https://doi.org/10.1137/0213003
  17. N. Megiddo, 'Linear-time algorithms for linear programming in and related problems,' In Proc. 23rd Annual IEEE Sym. Found. Comput. Sci., pp 329-338, 1982
  18. N. Megiddo, 'Linear programming in linear time when the dimension is fixed,' J. ACM, Vol. 31, No.1, pp.114-127, 1984 https://doi.org/10.1145/2422.322418
  19. R. Seidel, 'Small-dimensional linear programming and convex hulls made easy,' Discrete and Computational Geometry, Vol. 6, pp.423-434, 1991 https://doi.org/10.1007/BF02574699
  20. M. E. Hohmeyer, Implementation code of linear programming
  21. E. Welzl, 'Smallest enclosing disks (balls and ellipsoids),' New Results and New Trends in Computer Science, Springer Lecture Notes in Computer Science 555, pp.359-370, 1991 https://doi.org/10.1007/BFb0038202
  22. J. Erickson and H. Honda, Implementation code of the smallest enclosing sphere
  23. R. L. Graham, 'An efficient algorithm for determining the convex hull of finite planar set,' Information Processing Letters, Vol. 1, pp.132-133, 1972 https://doi.org/10.1016/0020-0190(72)90045-2
  24. J. O'Rourke, C. B. Chien, T. Olson, and D. Naddor, 'A new linear algorithm for intersecting convex polygons,' Computer Graphics and Image Processing, Vol. 19, pp.384-391, 1982 https://doi.org/10.1016/0146-664X(82)90023-5
  25. M. I. Shames and D. Hoey, 'Geometric inter-section problems,' Seventeenth Annual IEEE Symposium on Foundations of Computer Science, pp.208-215, 1976
  26. G. T. Toussaint, 'A simple linear algorithm for intersecting convex polygons,' The Visual Computer, Vol. 1, pp.118-123, 1985 https://doi.org/10.1007/BF01898355
  27. 하종성, '구상의 볼록 다각형의 교차 계산을 위한 선형 시간 알고리즘', 한국정보과학회지논문지:시스템및이론, Vol. 28, No.2, pp.58-63, 2001