Efficient Minimum Distance Computation for Solids of Revolution

We present a highly efficient algorithm for computing the minimum distance between two solids of revolution, each of which is defined by a planar cross-section region and a rotation axis. The boundary profile curve for the cross-section is first approx- imated by a bounding volume hierarchy (BVH) of fat arcs. By rotating the fat arcs around the axis, we generate the BVH of fat tori that bounds the surface of revolution. The minimum distance between two solids of revolution is then computed very efficiently using the distance between fat tori, which can be boiled down to the minimum distance computation for circles in the three-dimensional space. Our circle-based approach to the solids of revolution has distinctive features of geometric simplifica- tion. The main advantage is in the effectiveness of our approach in handling the complex cases where the minimum distance is obtained in non-convex regions of the solids under consideration. Though we are dealing with a geometric problem for solids, the algorithm actually works in a computational style similar to that of handling planar curves. Compared with conventional BVH-based methods, our algorithm demonstrates outperformance in computing speed, often 10–100 times faster. Moreover, the minimum distance can be computed very efficiently for the solids of revolution under deformation, where the dynamic reconstruction of fat arcs dominates the overall computation time and takes a few milliseconds

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

Surface-Surface-Intersection Computation using a Bounding Volume Hierarchy with Osculating Toroidal Patches in the Leaf Nodes

We present an efficient and robust algorithm for computing the intersection curve of two freeform surfaces using a Bounding Volume Hierarchy (BVH), where the leaf nodes contain osculating toroidal patches. The covering of each surface by a union of tightly fitting toroidal patches greatly simplifies the geometric operations involved in the surface-surface-intersection computation, i.e., the bounding of surface normals, the detection of surface binormals, the point projection from one surface to the other surface, and the intersection of local surface patches. Moreover, the hierarchy of simple bounding volumes (such as rectangle-swept spheres) accelerates the geometric search for the potential pairs of surface patches that may generate some curve segments in the surface-surface-intersection. We demonstrate the effectiveness of our approach by using test examples of intersecting two freeform surfaces, including some highly non-trivial examples with tangential intersections. In particular, we test the intersection of two almost identical surfaces, where one surface is obtained from the same surface, using a rotation around a normal line by a smaller and smaller angle θ = 10−k degree, k = 0, · · · , 5. The intersection results are often given as surface subpatches in some highly tangential areas, and even as the whole surface itself, when θ = 0.00001◦

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

Algorithms for cartographic visualization

Maps are effective tools for communicating information to the general public and help people to make decisions in, for example, navigation, spatial planning and politics. The mapmaker chooses the details to put on a map and the symbols to represent them. Not all details need to be geographic: thematic maps, which depict a single theme or attribute, such as population, income, crime rate, or migration, can very effectively communicate the spatial distribution of the visualized attribute. The vast amount of data currently available makes it infeasible to design all maps manually, and calls for automated cartography. In this thesis we presented efficient algorithms for the automated construction of various types of thematic maps. In Chapter 2 we studied the problem of drawing schematic maps. Schematic maps are a well-known cartographic tool; they visualize a set of nodes and edges (for example, highway or metro networks) in simplified form to communicate connectivity information as effectively as possible. Many schematic maps deviate substantially from the underlying geography since edges and vertices of the original network are moved in the simplification process. This can be a problem if we want to integrate the schematized network with a geographic map. In this scenario the schematized network has to be drawn with few orientations and links, while critical features (cities, lakes, etc.) of the base map are not obscured and retain their correct topological position with respect to the network. We developed an efficient algorithm to compute a collection of non-crossing paths with fixed orientations using as few links as possible. This algorithm approximates the optimal solution to within a factor that depends only on the number of allowed orientations. We can also draw the roads with different thicknesses, allowing us to visualize additional data related to the roads such as trafic volume. In Chapter 3 we studied methods to visualize quantitative data related to geographic regions. We first considered rectangular cartograms. Rectangular cartograms represent regions by rectangles; the positioning and adjacencies of these rectangles are chosen to suggest their geographic locations to the viewer, while their areas are chosen to represent the numeric values being communicated by the cartogram. One drawback of rectangular cartograms is that not every rectangular layout can be used to visualize all possible area assignments. Rectangular layouts that do have this property are called area-universal. We show that area-universal layouts are always one-sided, and we present algorithms to find one-sided layouts given a set of adjacencies. Rectangular cartograms often provide a nice visualization of quantitative data, but cartograms deform the underlying regions according to the data, which can make the map virtually unrecognizable if the data value differs greatly from the original area of a region or if data is not available at all for a particular region. A more direct method to visualize the data is to place circular symbols on the corresponding region, where the areas of the symbols correspond to the data. However, these maps, so-called symbol maps, can appear very cluttered with many overlapping symbols if large data values are associated with small regions. In Chapter 4 we proposed a novel type of quantitative thematic map, called necklace map, which overcomes these limitations. Instead of placing the symbols directly on a region, we place the symbols on a closed curve, the necklace, which surrounds the map. The location of a symbol on the necklace should be chosen in such a way that the relation between symbol and region is as clear as possible. Necklace maps appear clear and uncluttered and allow for comparatively large symbol sizes. We developed algorithms to compute necklace maps and demonstrated our method with experiments using various data sets and maps. In Chapter 5 and 6 we studied the automated creation of ow maps. Flow maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of ow between a source and a target. Good ow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding self-intersections. We developed a new algorithm for drawing ow trees, ow maps with a single source. Unlike existing methods, our method merges lines smoothly and avoids self-intersections. Our method is based on spiral trees, a new type of Steiner trees that we introduced. Spiral trees have an angle restriction which makes them appear smooth and hence suitable for drawing ow maps. We study the properties of spiral trees and give an approximation algorithm to compute them. We also show how to compute ow trees from spiral trees and we demonstrate our approach with extensive experiments

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

Abstracts for the twentyfirst European workshop on Computational geometry, Technische Universiteit Eindhoven, The Netherlands, March 9-11, 2005

This volume contains abstracts of the papers presented at the 21st European Workshop on Computational Geometry, held at TU Eindhoven (the Netherlands) on March 9–11, 2005. There were 53 papers presented at the Workshop, covering a wide range of topics. This record number shows that the field of computational geometry is very much alive in Europe. We wish to thank all the authors who submitted papers and presented their work at the workshop. We believe that this has lead to a collection of very interesting abstracts that are both enjoyable and informative for the reader. Finally, we are grateful to TU Eindhoven for their support in organizing the workshop and to the Netherlands Organisation for Scientific Research (NWO) for sponsoring the workshop

Safe planning and control via L1-adaptation and contraction theory

Autonomous robots that are capable of operating safely in the presence of imperfect model knowledge or external disturbances are vital in safety-critical applications. The research presented in this dissertation aims to enable safe planning and control for nonlinear systems with uncertainties using robust adaptive control theory. To this end we develop methods that (i) certify the collision-risk for the planned trajectories of autonomous robots, (ii) ensure guaranteed tracking performance in the presence of uncertainties, and (iii) learn the uncertainties in the model without sacrificing the transient performance guarantees, and (iv) learn incremental stability certificates parameterized as neural networks. 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 work, we present methods for a general class of absolutely continuous parametric curves to compute: the minimum separating distance, tolerance verification, and collision detection with respect to obstacles in the environment. A planning algorithm is incomplete if the robot is unable to safely track the planned trajectory. We introduce a feedback motion planning approach using contraction theory-based L1-adaptive (CL1) control to certify that planned trajectories of nonlinear systems with matched uncertainties are tracked with desired performance requirements. We present a planner-agnostic framework to design and certify invariant tubes around planned trajectories that the robot is always guaranteed to remain inside. By leveraging recent results in contraction analysis and L1-adaptive control we present an architecture that induces invariant tubes for nonlinear systems with state and time-varying uncertainties. Uncertainties caused by large modeling errors will significantly hinder the performance of any autonomous system. We adapt the CL1 framework to safely learn the uncertainties while simultaneously providing high-probability bounds on the tracking behavior. Any available data is incorporated into Gaussian process (GP) models of the uncertainties while the error in the learned model is quantified and handled by the CL1 controller to ensure that control objectives are met safely. As learning improves, so does the overall tracking performance of the system. This way, the safe operation of the system is always guaranteed, even during the learning transients. The tracking performance guarantees for nonlinear systems rely on the existence of incremental stability certificates that are prohibitively difficult to search for. We leverage the function approximation capabilities of deep neural networks for learning the certificates and the associated control policies jointly. The incremental stability properties of the closed-loop system are verified using interval arithmetic. The domain of the system is iteratively refined into a collection of intervals that certify the satisfaction of the stability properties over the interval regions. Thus, we avoid entirely rejecting the learned certificates and control policies just because they violate the stability properties in certain parts of the domain. We provide numerical experimentation on an inverted pendulum to validate our proposed methodology