최대 접촉 토러스 패치를 이용한 효율적인 기하학적 알고리즘

학위논문(석사) -- 서울대학교대학원 : 공과대학 컴퓨터공학부, 2021.8. 손상현.We present efficient geometric algorithms that are based upon toroidal patches. To begin with, we present to use osculating toroidal patches to approximate a regular surface and propose a reparametrization method for the approximating toroidal patches. Then, we show that the toroidal patches can approximate special kinds of freeform parametric surfaces that are built upon planar profil e curves much more effectively than general surfaces. Thanks to these precise toroidal patches, we can construct a very compact bounding volume hierarchy for a parametric surface. With the bounding volume hierarchy, we can perform fast and precise point projection, i.e., minimum distance computation from a point to the surface. Also, we can easily find binormal lines, i.e. lines that connect two geometric entities orthogonally, between toroidal patches and use them to find meaningful distance measures for parametric surfaces. We show that we can fi nd such binormal lines easily by fi nding binormal lines between circles in space. Using these fundamental toroidal geometric operations, we present an efficient minimum distance computation algorithm for solids of revolution. This algorithm accelerates the minimum distance computation 10-100 times faster than conventional method. Also, we propose an efficient Hausdorff Distance computation algorithm that is applicable to various kinds of parametric surfaces. We can fi nd the Hausdorff Distance, almost up to machine precision, without much cost increase. Even though these algorithms follow conventional frameworks in large, they exhibit much better precision and efficiency than previous methods because of the toroidal patches that we use in our hierarchy.본 논문에서는 토러스 패치를 이용한 효율적인 기하학적 알고리즘들을 소개한다. 먼저, 임의의 일반적인 정칙 곡면을 근사하기 위해 최대 접촉 토러스 패치를 사용할 것을 제안한다. 이를 위해 정칙 곡면의 변수를 토러스 패치의 변수로 변환하는 재매개화 공식을 제시한다. 이에 더해, 토러스 패치가 평면 곡선에 기반한 특수한 곡면들을 일반 곡면들보다 더 효과적으로 근사할 수 있음을 보인다. 이러한 토러스 패치의 정확성 덕분에, 임의의 곡면을 감싸는 굉장히 효율적인 bounding volume hierarchy를 얻을 수 있다. 이 자료 구조를 이용하여 공간 상의 한 점에서 해당 곡면으로의 점 투영 연산을 굉장히 빠르고 정확하게 할 수 있다. 또한, 곡면들 사이의 다양한 거리들을 찾기 위해 이 자료 구조에 저장된 토러스 패치들을 수직으로 연결하는 binormal 직선을 이용할 수 있다. 이러한 binormal 직선을 효율적으로 찾기 위해 공간 상의 원들을 이용할 수 있음을 보인다. 토러스 패치가 제공하는 위와 같은 기초적인 기하학적 연산들을 토대로, 효율적인 회전체 사이의 최단 거리 계산 알고리즘을 제시한다. 이 알고리즘은 기존의 알고리즘에 비해 10-100배 빠른 속도로 최단 거리를 계산한다. 또한, 효율적인 하우스도르프 거리 계산 알고리즘 역시 제안한다. 실험 결과, 이 알고리즘을 통해 거의 기계 정확도 내에서 정확한 하우스도르프 거리를 큰 비용 증가 없이 계산할 수 있었다. 이와 같은 성능 향상은 본 논문에서 사용한 토러스 패치의 정확성과 효율성에 기반하고 있다.Chapter 1 Introduction 1 1.1 Background 1 1.2 Research Objectives and Contributions 4 1.3 Thesis Organization 6 Chapter 2 Preliminaries 7 2.1 Freeform Parametric Surface 7 2.1.1 B ezier Surface 8 2.1.2 Surface of Revolution 9 2.1.3 Surface of Linear Extrusion 10 2.2 Torus 11 Chapter 3 Related Work 13 3.1 Bounding Volume Hierarchy 13 3.2 Minimum Distance Computation 15 3.3 Hausdor Distance Computation 15 Chapter 4 Bounding Volume Hierarchy 17 4.1 Construction 17 4.2 Toroidal Patch Approximation 19 4.2.1 Regular surface 19 4.2.2 Surface of Revolution 23 4.2.3 Surface of Linear Extrusion 24 4.3 Toroidal Operations 25 4.3.1 Point Projection 25 4.3.2 Binormal Computation 27 Chapter 5 Geometric Algorithms 30 5.1 Minimum distance computation for solids of revolution 30 5.1.1 General Framework 30 5.1.2 Algorithm 31 5.1.3 Experimental Results 33 5.2 Hausdor Distance computation 37 5.2.1 General Framework 37 5.2.2 Algorithm 39 5.2.3 Experimental Results 42 Chapter 6 Conculsion 50 Appendices 52 Chapter A Torus reparametrization 53 Bibliography 60 초록 67 Acknowledgments 68석

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