110 research outputs found

    Efficient Minimum Distance Computation for Solids of Revolution

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    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 10โ€“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

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    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

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    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 ฮธ = 10โˆ’k degree, k = 0, ยท ยท ยท , 5. The intersection results are often given as surface subpatches in some highly tangential areas, and even as the whole surface itself, when ฮธ = 0.00001โ—ฆ

    BVH์™€ ํ† ๋Ÿฌ์Šค ํŒจ์น˜๋ฅผ ์ด์šฉํ•œ ๊ณก๋ฉด ๊ต์ฐจ๊ณก์„  ์—ฐ์‚ฐ

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    ํ•™์œ„๋…ผ๋ฌธ(๋ฐ•์‚ฌ) -- ์„œ์šธ๋Œ€ํ•™๊ต๋Œ€ํ•™์› : ๊ณต๊ณผ๋Œ€ํ•™ ์ปดํ“จํ„ฐ๊ณตํ•™๋ถ€, 2021.8. ๊น€๋ช…์ˆ˜.๋‘ ๋ณ€์ˆ˜๋ฅผ ๊ฐ€์ง€๋Š” B-์Šคํ”Œ๋ผ์ธ ์ž์œ ๊ณก๋ฉด์˜ ๊ณก๋ฉด๊ฐ„ ๊ต์ฐจ๊ณก์„ ๊ณผ ์ž๊ฐ€ ๊ต์ฐจ๊ณก์„ , ๊ทธ๋ฆฌ๊ณ  ์˜คํ”„์…‹ ๊ณก๋ฉด์˜ ์ž๊ฐ€ ๊ต์ฐจ๊ณก์„ ์„ ๊ตฌํ•˜๋Š” ํšจ์œจ์ ์ด๊ณ  ์•ˆ์ •์ ์ธ ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ๊ฐœ๋ฐœํ•˜๋Š” ์ƒˆ๋กœ์šด ์ ‘๊ทผ ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•œ๋‹ค. ์ƒˆ๋กœ์šด ๋ฐฉ๋ฒ•์€ ์ตœํ•˜๋‹จ ๋…ธ๋“œ์— ์ตœ๋Œ€ ์ ‘์ด‰ ํ† ๋Ÿฌ์Šค๋ฅผ ๊ฐ€์ง€๋Š” ๋ณตํ•ฉ ๋ฐ”์šด๋”ฉ ๋ณผ๋ฅจ ๊ตฌ์กฐ์— ๊ธฐ๋ฐ˜์„ ๋‘๊ณ  ์žˆ๋‹ค. ์ด ๋ฐ”์šด๋”ฉ ๋ณผ๋ฅจ ๊ตฌ์กฐ๋Š” ๊ณก๋ฉด๊ฐ„ ๊ต์ฐจ๋‚˜ ์ž๊ฐ€ ๊ต์ฐจ๊ฐ€ ๋ฐœ์ƒํ•  ๊ฐ€๋Šฅ์„ฑ์ด ์žˆ๋Š” ์ž‘์€ ๊ณก๋ฉด ์กฐ๊ฐ ์Œ๋“ค์˜ ๊ธฐํ•˜ํ•™์  ๊ฒ€์ƒ‰์„ ๊ฐ€์†ํ™”ํ•œ๋‹ค. ์ตœ๋Œ€ ์ ‘์ด‰ ํ† ๋Ÿฌ์Šค๋Š” ์ž๊ธฐ๊ฐ€ ๊ทผ์‚ฌํ•œ C2-์—ฐ์† ์ž์œ ๊ณก๋ฉด๊ณผ 2์ฐจ ์ ‘์ด‰์„ ๊ฐ€์ง€๋ฏ€๋กœ ์ฃผ์–ด์ง„ ๊ณก๋ฉด์—์„œ ๋‹ค์–‘ํ•œ ๊ธฐํ•˜ ์—ฐ์‚ฐ์˜ ์ •๋ฐ€๋„๋ฅผ ํ–ฅ์ƒ์‹œํ‚ค๋Š”๋ฐ ํ•„์ˆ˜์ ์ธ ์—ญํ• ์„ ํ•œ๋‹ค. ํšจ์œจ์ ์ธ ๊ณก๋ฉด๊ฐ„ ๊ต์ฐจ๊ณก์„  ๊ณ„์‚ฐ์„ ์ง€์›ํ•˜๊ธฐ ์œ„ํ•ด, ๋ฏธ๋ฆฌ ๋งŒ๋“ค์–ด์ง„, ์ตœํ•˜๋‹จ ๋…ธ๋“œ์— ์ตœ๋Œ€ ์ ‘์ด‰ ํ† ๋Ÿฌ์Šค๊ฐ€ ์žˆ์œผ๋ฉฐ ๊ตฌํ˜•๊ตฌ๋ฉด ํŠธ๋ฆฌ๋ฅผ ๊ฐ€์ง€๋Š” ๋ณตํ•ฉ ์ดํ•ญ ๋ฐ”์šด๋”ฉ ๋ณผ๋ฅจ ๊ตฌ์กฐ๋ฅผ ์„ค๊ณ„ํ•˜์˜€๋‹ค. ์ตœ๋Œ€ ์ ‘์ด‰ ํ† ๋Ÿฌ์Šค๋Š” ๊ฑฐ์˜ ๋ชจ๋“  ๊ณณ์—์„œ ์ ‘์„ ๊ต์ฐจ๊ฐ€ ๋ฐœ์ƒํ•˜๋Š”, ์ž๋ช…ํ•˜์ง€ ์•Š์€ ๊ณก๋ฉด๊ฐ„ ๊ต์ฐจ๊ณก์„  ๊ณ„์‚ฐ ๋ฌธ์ œ์—์„œ๋„ ํšจ์œจ์ ์ด๊ณ  ์•ˆ์ •์ ์ธ ๊ฒฐ๊ณผ๋ฅผ ์ œ๊ณตํ•œ๋‹ค. ๊ณก๋ฉด์˜ ์ž๊ฐ€ ๊ต์ฐจ ๊ณก์„ ์„ ๊ตฌํ•˜๋Š” ๋ฌธ์ œ๋Š” ์ฃผ๋กœ ๋งˆ์ดํ„ฐ ์  ๋•Œ๋ฌธ์— ๊ณก๋ฉด๊ฐ„ ๊ต์ฐจ๊ณก์„ ์„ ๊ณ„์‚ฐํ•˜๋Š” ๊ฒƒ ๋ณด๋‹ค ํ›จ์”ฌ ๋” ์–ด๋ ต๋‹ค. ์ž๊ฐ€ ๊ต์ฐจ ๊ณก๋ฉด์€ ๋งˆ์ดํ„ฐ ์  ๋ถ€๊ทผ์—์„œ ๋ฒ•์„  ๋ฐฉํ–ฅ์ด ๊ธ‰๊ฒฉํžˆ ๋ณ€ํ•˜๋ฉฐ, ๋งˆ์ดํ„ฐ ์ ์€ ์ž๊ฐ€ ๊ต์ฐจ ๊ณก์„ ์˜ ๋์ ์— ์œ„์น˜ํ•œ๋‹ค. ๋”ฐ๋ผ์„œ ๋งˆ์ดํ„ฐ ์ ์€ ์ž๊ฐ€ ๊ต์ฐจ ๊ณก๋ฉด์˜ ๊ธฐํ•˜ ์—ฐ์‚ฐ ์•ˆ์ •์„ฑ์— ํฐ ๋ฌธ์ œ๋ฅผ ์ผ์œผํ‚จ๋‹ค. ๋งˆ์ดํ„ฐ ์ ์„ ์•ˆ์ •์ ์œผ๋กœ ๊ฐ์ง€ํ•˜์—ฌ ์ž๊ฐ€ ๊ต์ฐจ ๊ณก์„ ์˜ ๊ณ„์‚ฐ์„ ์šฉ์ดํ•˜๊ฒŒ ํ•˜๊ธฐ ์œ„ํ•ด, ์ž์œ ๊ณก๋ฉด์„ ์œ„ํ•œ ๋ณตํ•ฉ ๋ฐ”์šด๋”ฉ ๋ณผ๋ฅจ ๊ตฌ์กฐ์— ์ ์šฉํ•  ์ˆ˜ ์žˆ๋Š” ์‚ผํ•ญ ํŠธ๋ฆฌ ๊ตฌ์กฐ๋ฅผ ์ œ์‹œํ•œ๋‹ค. ํŠนํžˆ, ๋‘ ๋ณ€์ˆ˜๋ฅผ ๊ฐ€์ง€๋Š” ๊ณก๋ฉด์˜ ๋งค๊ฐœ๋ณ€์ˆ˜์˜์—ญ์—์„œ ๋งˆ์ดํ„ฐ ์ ์„ ์ถฉ๋ถ„ํžˆ ์ž‘์€ ์‚ฌ๊ฐํ˜•์œผ๋กœ ๊ฐ์‹ธ๋Š” ํŠน๋ณ„ํ•œ ํ‘œํ˜„ ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•œ๋‹ค. ์ ‘์„ ๊ต์ฐจ์™€ ๋งˆ์ดํ„ฐ ์ ์„ ๊ฐ€์ง€๋Š”, ์•„์ฃผ ์ž๋ช…ํ•˜์ง€ ์•Š์€ ์ž์œ ๊ณก๋ฉด ์˜ˆ์ œ๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ ์ƒˆ ๋ฐฉ๋ฒ•์ด ํšจ๊ณผ์ ์ž„์„ ์ž…์ฆํ•œ๋‹ค. ๋ชจ๋“  ์‹คํ—˜ ์˜ˆ์ œ์—์„œ, ๊ธฐํ•˜์š”์†Œ๋“ค์˜ ์ •ํ™•๋„๋Š” ํ•˜์šฐ์Šค๋„๋ฅดํ”„ ๊ฑฐ๋ฆฌ์˜ ์ƒํ•œ๋ณด๋‹ค ๋‚ฎ์Œ์„ ์ธก์ •ํ•˜์˜€๋‹ค.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 ์ดˆ๋ก 120๋ฐ•

    Algorithms for cartographic visualization

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    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

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    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

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    This volume contains abstracts of the papers presented at the 21st European Workshop on Computational Geometry, held at TU Eindhoven (the Netherlands) on March 9โ€“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

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    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
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