BVH와 토러스 패치를 이용한 곡면 교차곡선 연산

학위논문(박사) -- 서울대학교대학원 : 공과대학 컴퓨터공학부, 2021.8. 김명수.두 변수를 가지는 B-스플라인 자유곡면의 곡면간 교차곡선과 자가 교차곡선, 그리고 오프셋 곡면의 자가 교차곡선을 구하는 효율적이고 안정적인 알고리즘을 개발하는 새로운 접근 방법을 제시한다. 새로운 방법은 최하단 노드에 최대 접촉 토러스를 가지는 복합 바운딩 볼륨 구조에 기반을 두고 있다. 이 바운딩 볼륨 구조는 곡면간 교차나 자가 교차가 발생할 가능성이 있는 작은 곡면 조각 쌍들의 기하학적 검색을 가속화한다. 최대 접촉 토러스는 자기가 근사한 C2-연속 자유곡면과 2차 접촉을 가지므로 주어진 곡면에서 다양한 기하 연산의 정밀도를 향상시키는데 필수적인 역할을 한다. 효율적인 곡면간 교차곡선 계산을 지원하기 위해, 미리 만들어진, 최하단 노드에 최대 접촉 토러스가 있으며 구형구면 트리를 가지는 복합 이항 바운딩 볼륨 구조를 설계하였다. 최대 접촉 토러스는 거의 모든 곳에서 접선교차가 발생하는, 자명하지 않은 곡면간 교차곡선 계산 문제에서도 효율적이고 안정적인 결과를 제공한다. 곡면의 자가 교차 곡선을 구하는 문제는 주로 마이터 점 때문에 곡면간 교차곡선을 계산하는 것 보다 훨씬 더 어렵다. 자가 교차 곡면은 마이터 점 부근에서 법선 방향이 급격히 변하며, 마이터 점은 자가 교차 곡선의 끝점에 위치한다. 따라서 마이터 점은 자가 교차 곡면의 기하 연산 안정성에 큰 문제를 일으킨다. 마이터 점을 안정적으로 감지하여 자가 교차 곡선의 계산을 용이하게 하기 위해, 자유곡면을 위한 복합 바운딩 볼륨 구조에 적용할 수 있는 삼항 트리 구조를 제시한다. 특히, 두 변수를 가지는 곡면의 매개변수영역에서 마이터 점을 충분히 작은 사각형으로 감싸는 특별한 표현 방법을 제시한다. 접선교차와 마이터 점을 가지는, 아주 자명하지 않은 자유곡면 예제를 사용하여 새 방법이 효과적임을 입증한다. 모든 실험 예제에서, 기하요소들의 정확도는 하우스도르프 거리의 상한보다 낮음을 측정하였다.We present a new approach to the development of efficient and stable algorithms for intersecting freeform surfaces, including the surface-surface-intersection and the surface self-intersection of bivariate rational B-spline surfaces. Our new approach is based on a hybrid Bounding Volume Hierarchy(BVH) that stores osculating toroidal patches in the leaf nodes. The BVH structure accelerates the geometric search for the potential pairs of local surface patches that may intersect or self-intersect. Osculating toroidal patches have second-order contact with C2-continuous freeform surfaces that they approximate, which plays an essential role in improving the precision of various geometric operations on the given surfaces. To support efficient computation of the surface-surface-intersection curve, we design a hybrid binary BVH that is basically a pre-built Rectangle-Swept Sphere(RSS) tree enhanced with osculating toroidal patches in their leaf nodes. Osculating toroidal patches provide efficient and robust solutions to the problem even in the non-trivial cases of handling two freeform surfaces intersecting almost tangentially everywhere. The surface self-intersection problem is considerably more difficult than computing the intersection of two different surfaces, mainly due to the existence of miter points. A self-intersecting surface changes its normal direction dramatically around miter points, located at the open endpoints of the self-intersection curve. This undesirable behavior causes serious problems in the stability of geometric algorithms on self-intersecting surfaces. To facilitate surface self-intersection computation with a stable detection of miter points, we propose a ternary tree structure for the hybrid BVH of freeform surfaces. In particular, we propose a special representation of miter points using sufficiently small quadrangles in the parameter domain of bivariate surfaces and expand ideas to offset surfaces. We demonstrate the effectiveness of the proposed new approach using some highly non-trivial examples of freeform surfaces with tangential intersections and miter points. In all the test examples, the closeness of geometric entities is measured under the Hausdorff distance upper bound.Chapter 1 Introduction 1 1.1 Background 1 1.2 Surface-Surface-Intersection 5 1.3 Surface Self-Intersection 8 1.4 Main Contribution 12 1.5 Thesis Organization 14 Chapter 2 Preliminaries 15 2.1 Differential geometry of surfaces 15 2.2 Bezier curves and surfaces 17 2.3 Surface approximation 19 2.4 Torus 21 2.5 Summary 24 Chapter 3 Previous Work 25 3.1 Surface-Surface-Intersection 25 3.2 Surface Self-Intersection 29 3.3 Summary 32 Chapter 4 Bounding Volume Hierarchy for Surface Intersections 33 4.1 Binary Structure 33 4.1.1 Hierarchy of Bilinear Surfaces 34 4.1.2 Hierarchy of Planar Quadrangles 37 4.1.3 Construction of Leaf Nodes with Osculating Toroidal Patches 41 4.2 Ternary Structure 44 4.2.1 Miter Points 47 4.2.2 Leaf Nodes 50 4.2.3 Internal Nodes 51 4.3 Summary 56 Chapter 5 Surface-Surface-Intersection 57 5.1 BVH Traversal 58 5.2 Construction of SSI Curve Segments 59 5.2.1 Merging SSI Curve Segments with G1-Biarcs 60 5.2.2 Measuring the SSI Approximation Error Using G1-Biarcs 63 5.3 Tangential Intersection 64 5.4 Summary 65 Chapter 6 Surface Self-Intersection 67 6.1 Preprocessing 68 6.2 BVH Traversal 69 6.3 Construction of Intersection Curve Segments 70 6.4 Summary 72 Chapter 7 Trimming Offset Surfaces with Self-Intersection Curves 74 7.1 Offset Surface and Ternary Hybrid BVH 75 7.2 Preprocessing 77 7.3 Merging Intersection Curve Segments 81 7.4 Summary 84 Chapter 8 Experimental Results 85 8.1 Surface-Surface-Intersection 85 8.2 Surface Self-Intersection 97 8.2.1 Regular Surfaces 97 8.2.2 Offset Surfaces 100 Chapter 9 Conclusion 106 Bibliography 108 초록 120박

Interactive ray tracing of solvent excluded surfaces

Domain experts in fields concerned with the behavior of molecules, for example biochemists, employ simulations to study a molecule’s individual properties and mutual interactions with other molecules. To obtain an intuitive spatial understanding of the returned data of the simulations, various visualization techniques such as molecular surfaces can be applied on the data. The solvent excluded surface depicts the boundary between the molecule’s and a solvent’s occupied space and therefore the molecules accessibility for the solvent. Insight about a molecule’s potential for interaction such as reactions can be gained by studying the surface’s shape visually. Current implementations for the visualization of the surface usually utilize GPU ray casting to achieve the performance required to allow interactivity such as viewpoint changing. However, this makes implementation of physically motivated effects like ambient occlusion or global illumination difficult. If compute resources do not contain GPUs, which is often the case in compute clusters, expensive software rasterization has to be employed instead. As CPUs offer less parallelism compared to GPUs, overhead introduced by the overdraw of thousands of primitives should be avoided. To mitigate these issues, CPU visualization approaches resurfaced again in recent times. In this work, the solvent excluded surface is visualized interactively using the classic ray tracing approach within the OSPRay CPU ray tracing framework. The described implementation is able to compute and visualize the solvent excluded surface for datasets composed of millions of atoms. Additionally, the surface supports transparency rendering, which allows implementation of a cavity visualization method that uses ambient occlusion

Flickering nanometre-scale disorder in a crystal lattice tracked by plasmonic flare light emission

Abstract: The dynamic restructuring of metal nanoparticle surfaces is known to greatly influence their catalytic, electronic transport, and chemical binding functionalities. Here we show for the first time that non-equilibrium atomic-scale lattice defects can be detected in nanoparticles by purely optical means. These fluctuating states determine interface electronic transport for molecular electronics but because such rearrangements are low energy, measuring their rapid dynamics on single nanostructures by X-rays, electron beams, or tunnelling microscopies, is invasive and damaging. We utilise nano-optics at the sub-5nm scale to reveal rapid (on the millisecond timescale) evolution of defect morphologies on facets of gold nanoparticles on a mirror. Besides dynamic structural information, this highlights fundamental questions about defining bulk plasma frequencies for metals probed at the nanoscale

Flickering nanometre-scale disorder in a crystal lattice tracked by plasmonic flare light emission

Abstract: The dynamic restructuring of metal nanoparticle surfaces is known to greatly influence their catalytic, electronic transport, and chemical binding functionalities. Here we show for the first time that non-equilibrium atomic-scale lattice defects can be detected in nanoparticles by purely optical means. These fluctuating states determine interface electronic transport for molecular electronics but because such rearrangements are low energy, measuring their rapid dynamics on single nanostructures by X-rays, electron beams, or tunnelling microscopies, is invasive and damaging. We utilise nano-optics at the sub-5nm scale to reveal rapid (on the millisecond timescale) evolution of defect morphologies on facets of gold nanoparticles on a mirror. Besides dynamic structural information, this highlights fundamental questions about defining bulk plasma frequencies for metals probed at the nanoscale

Efficient numerical algorithms for surface formulations of mathematical models for biomolecule analysis and design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 179-183).This thesis presents a set of numerical techniques that extend and improve computational modeling approaches for biomolecule analysis and design. The presented research focuses on surface formulations of modeling problems related to the estimation of the energetic cost to transfer a biomolecule from the gas phase to aqueous solution. The thesis discusses four contributions to modeling biomolecular interactions. First, the thesis presents an approach to allow accurate discretization of the most prevalent mathematical definitions of the biomolecule-solvent interface; also presented are a number of accurate techniques for numerically integrating possibly singular functions over the discretized surfaces. Such techniques are essential for solving surface formulations numerically. The second part of the thesis presents a fast multiscale numerical algorithm, FFTSVD, that efficiently solves large boundary-element method problems in biomolecule electrostatics. The algorithm synthesizes elements of other popular fast algorithms to achieve excellent efficiency and flexibility. The third thesis component describes an integral-equation formulation and boundary-element method implementation for biomolecule electrostatic analysis.(cont.) The formulation and implementation allow the solution of complicated molecular topologies and physical models. Furthermore, by applying the methods developed in the first half of the thesis, the implementation can deliver superior accuracy for competitive performance. Finally, the thesis describes a highly efficient numerical method for calculating a biomolecular charge distribution that minimizes the free energy' change of binding to another molecule. The approach, which represents a novel PDE-constrained methodology, builds on well-developed physical theory. Computational results illustrate not only the method's improved performance but also its application to realistic biomolecule problems.by Jaydeep Porter Bardhan.Ph.D

Improvement of the survival in the seven-band grouper Epinephelus septemfasciatus larvae by optimizing aeration and water inlet in the mass-scale rearing tank

The water flow in larval rearing tanks has been indicated to cause mass mortality of the seven-band grouper Epinephelus septemfasciatus larvae. Therefore, a new aerating method was tested in an actual scale intensive rearing tank (8.0 m in diameter, 1.87 m of water depth, 100 m3 of volume), in which an aerator was positioned at the center of the rearing tank surrounding cylindrical drain (1.2 m in diameter) to generate the flow field, and seven larval rearing trials were performed. The survival rate with the former aeration methods were compared, in which several aerators were located in the rearing tank. The survival rate at 10 days after hatching with the new aeration method (61.5 ± 5.1%, n = 7) was approximately three times higher than the former methods (21.2 ± 13.7%, n = 6). The flow environment of rearing tanks was also examined by quantifying the flow field, and the relationship between the flow field in the rearing tank, behavior of larvae and survival discussed. It was confirmed that the vertical circulating flow was observed in rearing tanks, and determined effectively the survival and the behavior of grouper larvae in patchiness

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