본문 바로가기
KERD
논문KCI 등재

토러스 패치 기반의 정밀 근사를 이용한 자유곡면의 기하학적 처리

Geometric Processing for Freeform Surfaces Based on High-Precision Torus Patch Approximation

0

조회

0

다운로드

0

인용

0

참고문헌

키워드

초록

3차원상의 자유곡면에 대한 효율적인 기하 연산을 지원하기 위한 새로운 공간자료구조로서 토러스 패치 기반 정밀 근사를 이용한 자유곡면의 기하학적 처리 기법을 소개한다. 토러스는 곡률이 양이나 음인 경우뿐만 아니라, 0인 부분도 있으므로 자유곡면의 볼록, 오목 여부에 상관없이 곡면을 정밀하게 근사할 수 있다. 전통적인 기법과 달리 토러스 패치는 자유곡면의 법벡터 방향까지 쉽게 모델링할 수 있고, 토러스의 오프셋은 다시 토러스가 되므로 다양한 기하 연산의 가속화를 지원할 수 있다. 자유곡면과 이를 근사하는 토러스 패치 집합 사이의 양방향 하우스도르프 거 리의 상한을 계산하여 토러스 패치를 이용 하여 자유곡면을 높은 정밀도로 근사할 수 있음을 보였다. 이 기법을 이용하여 두 자유곡면의 교차곡선 계산과 자유곡면의 오프셋 곡면 생성을 쉽게 처리할 수 있음을 보였다.

Abstract

We introduce a geometric processing method for freeform surfaces based on high-precision torus patch approximation, a new spatial data structure for efficient geometric operations on freeform surfaces. A torus patch fits the freefonn surface with flexibility: it can handle not only positive and negative curvature but also a zero curvature. It is possible to precisely approximate the surface regardless of the convexity/concavity of the surface. Unlike the traditional method, a torus patch easily bounds the surface normal, and the offset of the torus becomes a torus again, thus helps the acceleration of various geometric operations. We have shown that the torus patch’s approximation accuracy of the freeform surface is high by measuring the upper bound of the two-sided Hausdorff distance between the freeform surface and set of torus patches. Using the method, it can be easily processed to detect an intersection curve between two freeform surfaces and find the offset surface of the freeform surface.

참고문헌 (41)

  1. S. Gottschalk, "OBBTree A Hierarchical Structure for Rapid Interference Detection", ACM Transactions on Graphics(SIGGRAPH 1996), 15(3), pp.171~180 (1996)
  2. Fast Proximity Queries with Swept Sphere Volumes, "E. Larsen", Dept. of Computer Science, UNC, ~ (1999)
  3. E. Larsen, "Fast distance queries with rectangular swept sphere volumes", Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No. 00CH37065), 4, pp.3719~3726 (2000)
  4. D. S. Meek, "Approximating smooth planar curves by arc splines", Journal of Computational and Applied Mathematics, 59(2), pp.221~231 (1995)
  5. M. Barton, "Spiral fat arcs-Bounding regions with cubic convergence", Graphical Models, 73(2), pp.50~57 (2011)
  6. Y.-J. Kim, "Data structures for accelerating geometric operations on freeform objects", Korea Computer Graphics Society 2011 Proceeding, pp.71~72 (2011)
  7. Y. -J. Kim, "Efficient offset trimming for planar rational curves using biarc trees", Computer Aided Geometric Design, 29(7), pp.555~564 (2012)
  8. J. Lee, "Efficient offset trimming for deformable planar curves using a dynamic hierarchy of bounding circular arcs", Computer-Aided Design, 58, pp.248~255 (2015)
  9. F. Aurenhammer, "Voronoi Diagrams — A Survey of a Fundamental Geometric Data Structure", ACM Computing Surveys(CSUR), 23(3), pp.345~405 (1991)
  10. O. Aichholzer, "Medial axis computation for planar free-form shapes", Computer-Aided Design, 41(5), pp.339~349 (2009)
  11. O. Aichholzer, "Divide-and-conquer for Voronoi diagrams revisited", Computational geometry, 43(8), pp.688~699 (2010)
  12. J. Lee, "Efficient voronoi diagram construction for planar freeform spiral curves", Computer Aided Geometric Design, 43, pp.131~142 (2016)
  13. S. Krishnan, "Rapid and Accurate Contact Determination between Spline Models using ShellTrees", Computer Graphics Forum, 17(3), pp.315~326 (1998)
  14. G. Bradshaw, "Sphere-Tree Construction using Dynamic Medial Axis Approximation", Proceedings of the 2002 ACM SIGGRAPH, 2002, 33~40
  15. X. -M. Liu, "A torus patch approximation approach for point projection on surfaces", Computer Aided Geometric Design, 26(5), pp.593~598 (2009)
  16. S.-M. Hu, "A second order algorithm for orthogonal projection onto curves and surfaces", Computer Aided Geometric Design, 22(3), pp.251~260 (2005)
  17. D. -S. Kim, "Tangent, normal, and visibility cones on B´ezier surfaces", Computer Aided Geometric Design, 12(3), pp.305~320 (1995)
  18. M. Daniel, "Using a Convex Pyramid to Bound Surface Normal Vectors", Computer graphics forum, 15(4), pp.219~227 (1996)
  19. K.-R. Park, "Offsets of ruled surfaces", Journal of the Korea Computer Graphics Society, 4(2), pp.69~75 (1998)
  20. R. T. Farouki, "The approximation of non-degenerate offset surfaces", Computer Aided Geometric Design, 3(1), pp.15~43 (1986)
  21. G. Elber, "Error bounded variable distance offset operator for free form curves and surfaces", International Journal of Computational Geometry & Applications, 1(01), pp.67~78 (1991)
  22. G. Elber, "Comparing offset curve approximation methods", IEEE computer graphics and applications, 17(3), pp.62~71 (1997)
  23. R. Martin, "Principal patches-a new class of surface patch based on differential geometry", Eurographics, 83, pp.47~55 (1983)
  24. R. T. Farouki, "Exact offset procedures for simple solids", Computer Aided Geometric Design, 2(4), pp.257~279 (1985)
  25. M. Tang, "Interactive hausdorff distance computation for general polygonal models", ACM Transactions on Graphics (TOG), 28(3), pp.74~ (2009)
  26. M. Bartoˇn, "Precise hausdorff distance computation between polygonal meshes", Computer Aided Geometric Design, 27(8), pp.580~591 (2010)
  27. Y. -J. Kim, "Precise Hausdorff distance computation for planar freeform curves using biarcs and depth buffer", The Visual Computer, 26(6-8), pp.1007~1016 (2010)
  28. M. Guthe, "Fast and accurate hausdorff distance calculation between meshes", Journal of WSCG, 13(2) (2005)
  29. Y. Kang, "Fast and robust Hausdorff distance computation from triangle mesh to quad mesh in near-zero cases", Computer Aided Geometric Design, 62, pp.91~103 (2018)
  30. I. Hanniel, "Computing the Hausdorff distance between NURBS surfaces using numerical iteration on the GPU", Graphical Models, 74(4), pp.255~264 (2012)
  31. A. Krishnamurthy, "GPUaccelerated Hausdorff distance computation between dynamic deformable NURBS surfaces", Computer-Aided Design, 43(11), pp.1370~1379 (2011)
  32. Y. -J. Kim, "Efficient Hausdorff distance computation for freeform geometric models in close proximity", Computer-Aided Design, 45(2), pp.270~276 (2013)
  33. D. Filip, "Surface algorithms using bounds on derivatives", Computer Aided Geometric Design, 3(4), pp.295~311 (1986)
  34. M. P. Do Carmo, "Differential Geometry of Curves and Surfaces" (2016)
  35. Y. Kang, "Deformable quad mesh for accelerated geometric operations", Korea Computer Graphics Society 2015 Proceeding, pp.73~74 (2011)
  36. P. Du, "TSS BVHs : tetrahedron swept sphere BVHs for ray tracing subdivision surfaces", Computer Graphics Forum, 35(7), pp.279~288 (2016)
  37. N. M. Patrikalakis, "Surface-to-surface intersections", IEEE Computer Graphics and Applications, 13(1), pp.89~95 (1993)
  38. K. -J. Kim, "Computing intersection between freeform surfaces", Journal of the Korea Computer Graphics Society, 10(3), pp.28~33 (2004)
  39. K. -J. Kim, "Torus", Sphere Intersection Based on a Configuration Space Approach, 1998, 77~92
  40. K. -J. Kim, "Circles in torus-torus intersections", Journal of Computational and Applied Mathematics, 236(9), pp.2387~2397 (2012)
  41. G. Elber, "Hausdorff and Minimal Distances between Parametric Freeforms in R2 and R3", International Conference on Geometric Modeling and Processing, pp.191~204 (2008)

관련 논문AI추천

이 간행물의 다른 논문 (14건)