Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry

Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the present paper is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently rather separated domains: Smooth Dynamics, Diophantine Geometry, Approximation Theory, and Computational Geometry. The structure of the results, open problems, and conjectures in each of these domains shows in many cases a remarkable similarity, which we try to stress. Sometimes this similarity can be easily explained, sometimes the reasons remain somewhat obscure, and it motivates some natural questions discussed in the paper. We present also some new results, stressing interconnection between various types and various applications of smooth parametrization

오프셋 곡선 및 곡면의 자가 교차 검출 및 제거

학위논문(박사)--서울대학교 대학원 :공과대학 컴퓨터공학부,2020. 2. 김명수.Offset curves and surfaces have many applications in computer-aided design and manufacturing, but the self-intersections and redundancies must be trimmed away for their practical use. We present a new method for offset curve and surface trimming that detects the self-intersections and eliminates the redundant parts of an offset curve and surface that are closer than the offset distance to the original curve and surface. We first propose an offset trimming method based on constructing geometric constraint equations. We formulate the constraint equations of the self-intersections of an offset curve and surface in the parameter domain of the original curve and surface. Numerical computations based on the regularity and intrinsic properties of the given input curve and surface is carried out to compute the solution of the constraint equations. The method deals with numerical instability around near-singular regions of an offset surface by using osculating tori that can be constructed in a highly stable way, i.e., by offsetting the osculating torii of the given input regular surface. We reveal the branching structure and the terminal points from the complete self-intersection curves of the offset surface. From the observation that the trimming method based on the multivariate equation solving is computationally expensive, we also propose an acceleration technique to trim an offset curve and surface. The alternative method constructs a bounding volume hierarchy specially designed to enclose the offset curve and surface and detects the self-collision of the bounding volumes instead. In the case of an offset surface, the thickness of the bounding volumes is indirectly determined based on the maximum deviations of the positions and the normals between the given input surface patches and their osculating tori. For further acceleration, the bounding volumes are pruned as much as possible during self-collision detection using various geometric constraints imposed on the offset surface. We demonstrate the effectiveness of the new trimming method using several non-trivial test examples of offset trimming. Lastly, we investigate the problem of computing the Voronoi diagram of a freeform surface using the offset trimming technique for surfaces. By trimming the offset surface with a gradually changing offset radius, we compute the boundary of the Voronoi cells that appear in the concave side of the given input surface. In particular, we interpret the singular and branching points of the self-intersection curves of the trimmed offset surfaces in terms of the boundary elements of the Voronoi diagram.오프셋 곡선 및 곡면은 computer-aided design (CAD)와 computer-aided manufacturing (CAM)에서 널리 이용되는 연산들 중 하나이다. 하지만 실용적인 활용을 위해서는 오프셋 곡선 및 곡면에서 생기는 자가 교차를 찾고 이를 기준으로 오프셋 곡선 및 곡면에서 원래의 곡선 및 곡면에 가까운 불필요한 영역을 제거하여야한다. 본 논문에서는 오프셋 곡선 및 곡면에서 생기는 자가 교차를 계산하고, 오프셋 곡선 및 곡면에서 생기는 불필요한 영역을 제거하는 알고리즘을 제안한다. 본 논문은 우선 오프셋 곡선 및 곡면의 자가 교차점들과 그 교차점들이 기인한 원래 곡선 및 곡면의 점들이 이루는 평면 이등변 삼각형 관계로부터 오프셋 곡선 및 곡면의 자가 교차점의 제약 조건을 만족시키는 방정식들을 세운다. 이 제약식들은 원래 곡선 및 곡면의 변수 공간에서 표현되며, 이 방정식들의 해는 다변수 방정식의 해를 구하는 solver를 이용하여 구한다. 오프셋 곡면의 경우, 원래 곡면의 주곡률 중 하나가 오프셋 반지름의 역수와 같을 때 오프셋 곡면의 법선이 정의가 되지 않는 특이점이 생기는데, 오프셋 곡면의 자가 교차 곡선이 이 부근을 지날 때는 자가 교차 곡선의 계산이 불안정해진다. 따라서 자가 교차 곡선이 오프셋 곡면의 특이점 부근을 지날 때는 오프셋 곡면을 접촉 토러스로 치환하여 더 안정된 방법으로 자가 교차 곡선을 구한다. 계산된 오프셋 곡면의 자가 교차 곡선으로부터 교차 곡선의 $xyz$-공간에서의 말단 점, 가지 구조 등을 밝힌다. 본 논문은 또한 바운딩 볼륨 기반의 오프셋 곡선 및 곡면의 자가 교차 곡선 검출을 가속화하는 방법을 제시한다. 바운딩 볼륨은 기저 곡선 및 곡면을 단순한 기하로 감싸고 기하 연산을 수행함으로써 가속화에 기여한다. 오프셋 곡면의 자가 교차 곡선을 구하기 위하여, 본 논문은 오프셋 곡면의 바운딩 볼륨 구조를 기저 곡면의 바운딩 볼륨과 기저 곡면의 법선 곡면의 바운딩 볼륨의 구조로부터 계산하며 이때 각 바운딩 볼륨의 두께를 계산한다. 또한, 바운딩 볼륨 중에서 실제 오프셋 곡선 및 곡면의 자가 교차에 기여하지 않는 부분을 깊은 재귀 전에 찾아서 제거하는 여러 조건들을 나열한다. 한편, 자가 교차가 제거된 오프셋 곡선 및 곡면은 기저 곡선 및 곡면의 보로노이 구조와 깊은 관련이 있는 것이 알려져 있다. 본 논문에서는 자유 곡면의 연속된 오프셋 곡면들로부터 자유 곡면의 보로노이 구조를 유추하는 방법을 제시한다. 특히, 오프셋 곡면의 자가 교차 곡선 상에서 나타나는 가지 점이나 말단 점과 같은 특이점들이 자유 곡면의 보로노이 구조에서 어떻게 해석되는지 제시한다.1. Introduction 1 1.1 Background and Motivation 1 1.2 Research Objectives and Approach 7 1.3 Contributions and Thesis Organization 11 2. Preliminaries 14 2.1 Curve and Surface Representation 14 2.1.1 Bezier Representation 14 2.1.2 B-spline Representation 17 2.2 Differential Geometry of Curves and Surfaces 19 2.2.1 Differential Geometry of Curves 19 2.2.2 Differential Geometry of Surfaces 21 3. Previous Work 23 3.1 Offset Curves 24 3.2 Offset Surfaces 27 3.3 Offset Curves on Surfaces 29 4. Trimming Offset Curve Self-intersections 32 4.1 Experimental Results 35 5. Trimming Offset Surface Self-intersections 38 5.1 Constraint Equations for Offset Self-Intersections 38 5.1.1 Coplanarity Constraint 39 5.1.2 Equi-angle Constraint 40 5.2 Removing Trivial Solutions 40 5.3 Removing Normal Flips 41 5.4 Multivariate Solver for Constraints 43 5.A Derivation of f(u,v) 46 5.B Relationship between f(u,v) and Curvatures 47 5.3 Trimming Offset Surfaces 50 5.4 Experimental Results 53 5.5 Summary 57 6. Acceleration of trimming offset curves and surfaces 62 6.1 Motivation 62 6.2 Basic Approach 67 6.3 Trimming an Offset Curve using the BVH 70 6.4 Trimming an Offset Surface using the BVH 75 6.4.1 Offset Surface BVH 75 6.4.2 Finding Self-intersections in Offset Surface Using BVH 87 6.4.3 Tracing Self-intersection Curves 98 6.5 Experimental Results 100 6.6 Summary 106 7. Application of Trimming Offset Surfaces: 3D Voronoi Diagram 107 7.1 Background 107 7.2 Approach 110 7.3 Experimental Results 112 7.4 Summary 114 8. Conclusion 119 Bibliography iDocto

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