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

Proximity Queries for Absolutely Continuous Parametric Curves

In motion planning problems for autonomous robots, such as self-driving cars, the robot must ensure that its planned path is not in close proximity to obstacles in the environment. However, the problem of evaluating the proximity is generally non-convex and serves as a significant computational bottleneck for motion planning algorithms. In this paper, we present methods for a general class of absolutely continuous parametric curves to compute: (i) the minimum separating distance, (ii) tolerance verification, and (iii) collision detection. Our methods efficiently compute bounds on obstacle proximity by bounding the curve in a convex region. This bound is based on an upper bound on the curve arc length that can be expressed in closed form for a useful class of parametric curves including curves with trigonometric or polynomial bases. We demonstrate the computational efficiency and accuracy of our approach through numerical simulations of several proximity problems.Comment: Proceedings of Robotics: Science and System

High-Quality Simplification and Repair of Polygonal Models

Because of the rapid evolution of 3D acquisition and modelling methods, highly complex and detailed polygonal models with constantly increasing polygon count are used as three-dimensional geometric representations of objects in computer graphics and engineering applications. The fact that this particular representation is arguably the most widespread one is due to its simplicity, flexibility and rendering support by 3D graphics hardware. Polygonal models are used for rendering of objects in a broad range of disciplines like medical imaging, scientific visualization, computer aided design, film industry, etc. The handling of huge scenes composed of these high-resolution models rapidly approaches the computational capabilities of any graphics accelerator. In order to be able to cope with the complexity and to build level-of-detail representations, concentrated efforts were dedicated in the recent years to the development of new mesh simplification methods that produce high-quality approximations of complex models by reducing the number of polygons used in the surface while keeping the overall shape, volume and boundaries preserved as much as possible. Many well-established methods and applications require "well-behaved" models as input. Degenerate or incorectly oriented faces, T-joints, cracks and holes are just a few of the possible degenaracies that are often disallowed by various algorithms. Unfortunately, it is all too common to find polygonal models that contain, due to incorrect modelling or acquisition, such artefacts. Applications that may require "clean" models include finite element analysis, surface smoothing, model simplification, stereo lithography. Mesh repair is the task of removing artefacts from a polygonal model in order to produce an output model that is suitable for further processing by methods and applications that have certain quality requirements on their input. This thesis introduces a set of new algorithms that address several particular aspects of mesh repair and mesh simplification. One of the two mesh repair methods is dealing with the inconsistency of normal orientation, while another one, removes the inconsistency of vertex connectivity. Of the three mesh simplification approaches presented here, the first one attempts to simplify polygonal models with the highest possible quality, the second, applies the developed technique to out-of-core simplification, and the third, prevents self-intersections of the model surface that can occur during mesh simplification

From 3D Models to 3D Prints: an Overview of the Processing Pipeline

Due to the wide diffusion of 3D printing technologies, geometric algorithms for Additive Manufacturing are being invented at an impressive speed. Each single step, in particular along the Process Planning pipeline, can now count on dozens of methods that prepare the 3D model for fabrication, while analysing and optimizing geometry and machine instructions for various objectives. This report provides a classification of this huge state of the art, and elicits the relation between each single algorithm and a list of desirable objectives during Process Planning. The objectives themselves are listed and discussed, along with possible needs for tradeoffs. Additive Manufacturing technologies are broadly categorized to explicitly relate classes of devices and supported features. Finally, this report offers an analysis of the state of the art while discussing open and challenging problems from both an academic and an industrial perspective.Comment: European Union (EU); Horizon 2020; H2020-FoF-2015; RIA - Research and Innovation action; Grant agreement N. 68044

On initialization of milling paths for 5-axis flank CNC machining of free-form surfaces with general milling tools

We propose a path-planning algorithm for 5-axis flank CNC machining with general tools of varying curvature. Our approach generalizes the initialization strategy introduced for conical tools [Bo et al., 2017] to arbitrary milling tools. Given a free-form (NURBS) surface and a rotational milling tool, we look for its motion in 3D to approximate the input reference surface within a given tolerance. We show that for a general shape of the milling tool, there exist locally and generically four 3D directions in which the point-surface distance follows the shape of the tool up to second order. These directions form a 3D multi-valued vector field and its integration gives rise to a set of integral curves. Among these integral curves, we seek straight line segments that correspond to good initial positions of the axes of the milling tool. We validate our method against synthetic examples with known exact solutions and, on industrial datasets, we detect approximate solutions that meet fine machining tolerances. We also demonstrate applicability of our method for efficient flank milling of convex regions that is not possible using traditional conical tools.RYC-2017-2264

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

Indexing 3D scenes using the interaction bisector surface

The spatial relationship between different objects plays an important role in defining the context of scenes. Most previous 3D classification and retrieval methods take into account either the individual geometry of the objects or simple relationships between them such as the contacts or adjacencies. In this article we propose a new method for the classification and retrieval of 3D objects based on the Interaction Bisector Surface (IBS), a subset of the Voronoi diagram defined between objects. The IBS is a sophisticated representation that describes topological relationships such as whether an object is wrapped in, linked to, or tangled with others, as well as geometric relationships such as the distance between objects. We propose a hierarchical framework to index scenes by examining both the topological structure and the geometric attributes of the IBS. The topology-based indexing can compare spatial relations without being severely affected by local geometric details of the object. Geometric attributes can also be applied in comparing the precise way in which the objects are interacting with one another. Experimental results show that our method is effective at relationship classification and content-based relationship retrieval

IST Austria Thesis

Fabrication of curved shells plays an important role in modern design, industry, and science. Among their remarkable properties are, for example, aesthetics of organic shapes, ability to evenly distribute loads, or efficient flow separation. They find applications across vast length scales ranging from sky-scraper architecture to microscopic devices. But, at the same time, the design of curved shells and their manufacturing process pose a variety of challenges. In this thesis, they are addressed from several perspectives. In particular, this thesis presents approaches based on the transformation of initially flat sheets into the target curved surfaces. This involves problems of interactive design of shells with nontrivial mechanical constraints, inverse design of complex structural materials, and data-driven modeling of delicate and time-dependent physical properties. At the same time, two newly-developed self-morphing mechanisms targeting flat-to-curved transformation are presented. In architecture, doubly curved surfaces can be realized as cold bent glass panelizations. Originally flat glass panels are bent into frames and remain stressed. This is a cost-efficient fabrication approach compared to hot bending, when glass panels are shaped plastically. However such constructions are prone to breaking during bending, and it is highly nontrivial to navigate the design space, keeping the panels fabricable and aesthetically pleasing at the same time. We introduce an interactive design system for cold bent glass façades, while previously even offline optimization for such scenarios has not been sufficiently developed. Our method is based on a deep learning approach providing quick and high precision estimation of glass panel shape and stress while handling the shape multimodality. Fabrication of smaller objects of scales below 1 m, can also greatly benefit from shaping originally flat sheets. In this respect, we designed new self-morphing shell mechanisms transforming from an initial flat state to a doubly curved state with high precision and detail. Our so-called CurveUps demonstrate the encodement of the geometric information into the shell. Furthermore, we explored the frontiers of programmable materials and showed how temporal information can additionally be encoded into a flat shell. This allows prescribing deformation sequences for doubly curved surfaces and, thus, facilitates self-collision avoidance enabling complex shapes and functionalities otherwise impossible. Both of these methods include inverse design tools keeping the user in the design loop

PHYSICS-AWARE MODEL SIMPLIFICATION FOR INTERACTIVE VIRTUAL ENVIRONMENTS

Rigid body simulation is an integral part of Virtual Environments (VE) for autonomous planning, training, and design tasks. The underlying physics-based simulation of VE must be accurate and computationally fast enough for the intended application, which unfortunately are conflicting requirements. Two ways to perform fast and high fidelity physics-based simulation are: (1) model simplification, and (2) parallel computation. Model simplification can be used to allow simulation at an interactive rate while introducing an acceptable level of error. Currently, manual model simplification is the most common way of performing simulation speedup but it is time consuming. Hence, in order to reduce the development time of VEs, automated model simplification is needed. The dissertation presents an automated model simplification approach based on geometric reasoning, spatial decomposition, and temporal coherence. Geometric reasoning is used to develop an accessibility based algorithm for removing portions of geometric models that do not play any role in rigid body to rigid body interaction simulation. Removing such inaccessible portions of the interacting rigid body models has no influence on the simulation accuracy but reduces computation time significantly. Spatial decomposition is used to develop a clustering algorithm that reduces the number of fluid pressure computations resulting in significant speedup of rigid body and fluid interaction simulation. Temporal coherence algorithm reuses the computed force values from rigid body to fluid interaction based on the coherence of fluid surrounding the rigid body. The simulations are further sped up by performing computing on graphics processing unit (GPU). The dissertation also presents the issues pertaining to the development of parallel algorithms for rigid body simulations both on multi-core processors and GPU. The developed algorithms have enabled real-time, high fidelity, six degrees of freedom, and time domain simulation of unmanned sea surface vehicles (USSV) and can be used for autonomous motion planning, tele-operation, and learning from demonstration applications