Showing papers on "Computational geometry published in 2015"

Four-Dimensional Printing for Freeform Surfaces: Design Optimization of Origami and Kirigami Structures

Abstract: A self-folding structure fabricated by additive manufacturing (AM) can be automatically folded into a demanding three-dimensional (3D) shape by actuation mechanisms such as heating. However, 3D surfaces can only be fabricated by self-folding structures when they are flattenable. Most generally, designed parts are not flattenable. To address the problem, we develop a shape optimization method to modify a nonflattenable surface into flattenable. The shape optimization framework is equipped with topological operators for adding interior/boundary cuts to further improve the flattenability. When inserting cuts, self-intersection is locally prevented on the flattened two-dimensional (2D) pieces. The total length of inserted cuts is also minimized to reduce artifacts on the finally folded 3D shape. [DOI: 10.1115/1.4031023]

CGALmesh: A Generic Framework for Delaunay Mesh Generation

Abstract: CGALmesh is the mesh generation software package of the Computational Geometry Algorithm Library (CGAL). It generates isotropic simplicial meshes—surface triangular meshes or volume tetrahedral meshes—from input surfaces, 3D domains, and 3D multidomains, with or without sharp features. The underlying meshing algorithm relies on restricted Delaunay triangulations to approximate domains and surfaces and on Delaunay refinement to ensure both approximation accuracy and mesh quality. CGALmesh provides guarantees on approximation quality and on the size and shape of the mesh elements. It provides four optional mesh optimization algorithms to further improve the mesh quality. A distinctive property of CGALmesh is its high flexibility with respect to the input domain representation. Such a flexibility is achieved through a careful software design, gathering into a single abstract concept, denoted by the oracle, all required interface features between the meshing engine and the input domain. We already provide oracles for domains defined by polyhedral and implicit surfaces.

Approaches for the assembly simulation of skin model shapes

Abstract: Even though they are weakly noticed, geometric part deviations accompany our everyday life These geometric deviations affect the assemblability and functional compliance of products, since small part variations accumulate through large-scale assemblies and lead to malfunction as well as decreased product reliability and safety However, the consideration of part deviations in the virtual modelling of mechanical assemblies is an ongoing challenge in computer-aided tolerancing research This is because the resulting assembly configurations for variant parts are far more complicated than for nominal assemblies In this contribution, two approaches for the relative positioning of point based models are highlighted and adapted to the assembly simulation of Skin Model Shapes, which are specific workpiece representatives considering geometric deviations The first approach employs constrained registration techniques to determine the position of variant parts in an assembly considering multiple assembly steps simultaneously, whereas the second utilizes the difference surface to solve the positioning problem sequentially The application of these approaches to computer-aided tolerancing is demonstrated, though their applicability reaches various fields of industrial geometry Skin Model Shapes are digital part representatives comprising geometric deviationsApproaches for the relative positioning of point-based Skin Model Shapes are proposedThe approaches ground on algorithms from computational geometry and computer graphicsApplications for the assembly simulation in tolerancing are given

Helly's Theorem: New Variations and Applications

Abstract: This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in optimization.

Fast Wavefront Propagation (FWP) for Computing Exact Geodesic Distances on Meshes

Abstract: Computing geodesic distances on triangle meshes is a fundamental problem in computational geometry and computer graphics. To date, two notable classes of algorithms, the Mitchell-Mount-Papadimitriou (MMP) algorithm and the Chen-Han (CH) algorithm, have been proposed. Although these algorithms can compute exact geodesic distances if numerical computation is exact, they are computationally expensive, which diminishes their usefulness for large-scale models and/or time-critical applications. In this paper, we propose the fast wavefront propagation (FWP) framework for improving the performance of both the MMP and CH algorithms. Unlike the original algorithms that propagate only a single window (a data structure locally encodes geodesic information) at each iteration, our method organizes windows with a bucket data structure so that it can process a large number of windows simultaneously without compromising wavefront quality. Thanks to its macro nature, the FWP method is less sensitive to mesh triangulation than the MMP and CH algorithms. We evaluate our FWP-based MMP and CH algorithms on a wide range of large-scale real-world models. Computational results show that our method can improve the speed by a factor of 3-10.

Instance Optimal Geometric Algorithms

Abstract: We prove the existence of an algorithm $A$ for computing 2-d or 3-d convex hulls that is optimal for every point set in the following sense: for every sequence $\sigma$ of $n$ points and for every algorithm $A'$ in a certain class $\mathcal{A}$, the running time of $A$ on input $\sigma$ is at most a constant factor times the maximum running time of $A'$ on the worst possible permutation of $\sigma$ for $A'$. We establish a stronger property: for every sequence $\sigma$ of points and every algorithm $A'$, the running time of $A$ on $\sigma$ is at most a constant factor times the average running time of $A'$ over all permutations of $\sigma$. We call algorithms satisfying these properties instance-optimal in the order-oblivious and random-order setting. Such instance-optimal algorithms simultaneously subsume output-sensitive algorithms and distribution-dependent average-case algorithms, and all algorithms that do not take advantage of the order of the input or that assume the input is given in a random order. The class $\mathcal{A}$ under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments. To establish an instance-specific lower bound, we deviate from traditional Ben-Or-style proofs and adopt a new adversary argument. For 2-d convex hulls, we prove that a version of the well known algorithm by Kirkpatrick and Seidel (1986) or Chan, Snoeyink, and Yap (1995) already attains this lower bound. For 3-d convex hulls, we propose a new algorithm. We further obtain instance-optimal results for a few other standard problems in computational geometry. Our framework also reveals connection to distribution-sensitive data structures and yields new results as a byproduct, for example, on on-line orthogonal range searching in 2-d and on-line halfspace range reporting in 2-d and 3-d.

Computational Geometry Column 62

Abstract: In this column, we consider natural problems in computational geometry that are polynomialtime equivalent to finding a real solution to a system of polynomial inequalities. Such problems are called ⇿R-complete, and typically involve geometric graphs. We describe the foundations of those completeness proofs, in particular Mnev's Universality Theorem, as well as some known ⇿R-completeness results, and recent additions to the list. The results shed light on the complex structure of those problems, beyond mere NP-hardness.

Constrained Reachability and Controllability Sets for Planetary Precision Landing via Convex Optimization

Abstract: This paper presents a convex optimizations-based method to compute the set of initial conditions from which a given landing accuracy to a target can be achieved (constrained controllability set) and the set of states that can be reached from a given set of initial states (constrained reachability set) for a planetary landing vehicle with all the relevant control and mission constraints. The proposed method is based on the lossless convexification of the powered-descent landing guidance problem and methods of convex optimization and computational geometry. These techniques are used to generate approximations that can be arbitrarily close to the actual reachability or controllability sets. The quantification of these sets allows evaluation of the feasibility of a prescribed landing accuracy for a given vehicle and an expected set of dispersions from the parachute descent phase of a planetary landing mission. Since these sets are generated systematically and quickly, a wide range of design options can be eva...

ConformalALU: A Conformal Geometric Algebra Coprocessor for Medical Image Processing

Abstract: Medical imaging involves important computational geometric problems, such as image segmentation and analysis, shape approximation, three-dimensional (3D) modeling, and registration of volumetric data. In the last few years, Conformal Geometric Algebra (CGA), based on five-dimensional (5D) Clifford Algebra, is emerging as a new paradigm that offers simple and universal operators for the representation and solution of complex geometric problems. However, the widespread use of CGA has been so far hindered by its high dimensionality and computational complexity. This paper proposes a simplified formulation of the conformal geometric operations (reflections, rotations, translations, and uniform scaling) aimed at a parallel hardware implementation. A specialized coprocessing architecture (ConformalALU) that offers direct hardware support to the new CGA operators, is also presented. The ConformalALU has been prototyped as a complete System-on-Programmable-Chip (SoPC) on the Xilinx ML507 FPGA board, containing a Virtex-5 FPGA device. Experimental results show average speedups of one order of magnitude for CGA rotations, translations, and dilations with respect to the geometric algebra software library Gaigen running on the general-purpose PowerPC processor embedded in the target FPGA device. A suite of medical imaging applications, including segmentation, 3D modeling and registration of medical data, has been used as testbench to evaluate the coprocessor effectiveness.

Closed-form characterization of the Minkowski sum and difference of two ellipsoids

Abstract: This paper makes three original contributions: (1) Explicit closed-form parametric formulas for the boundary of the Minkowski sum and difference of two arbitrarily oriented solid ellipsoids in n-dimensional Euclidean space are presented; (2) Based on this, new closed-form lower and upper bounds for the volume contained in these Minkowski sums and differences are derived in the 2D and 3D cases and these bounds are shown to be better than those in the existing literature; (3) A demonstration of how these ideas can be applied to problems in computational geometry and robotics is provided, and a relationship to the Principal Kinematic Formula from the fields of integral geometry and geometric probability is uncovered.

Map Construction Algorithms

Abstract: The book provides an overview of the state-of-the-art of map construction algorithms, which use tracking data in the form of trajectories to generate vector maps. The most common trajectory type is GPS-based trajectories. It introduces three emerging algorithmic categories, outlines their general algorithmic ideas, and discusses three representative algorithms in greater detail. To quantify map construction algorithms, the authors include specific datasets and evaluation measures. The datasets, source code of map construction algorithms and evaluation measures are publicly available on http://www.mapconstruction.org. The web site serves as a repository for map construction data and algorithms and researchers can contribute by uploading their own code and benchmark data. Map Construction Algorithms is an excellent resource for professionals working in computational geometry, spatial databases, and GIS. Advanced-level students studying computer science, geography and mathematics will also find this book a useful tool.

Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

Abstract: We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (1992) and the optimal but randomized algorithm of Ramos (1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, (<= k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, epsilon-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (1991) and Chazelle (1993).

Visibility and ray shooting queries in polygonal domains

Abstract: Given a polygonal domain (or polygon with holes) in the plane, we study the problem of computing the visibility polygon of any query point. As a special case of visibility problems, we also study the ray-shooting problem of finding the first point on the polygon boundaries that is hit by any query ray. These are fundamental problems in computational geometry and have been studied extensively. We present new algorithms and data structures that improve the previous results.

L-Visibility Drawings of IC-Planar Graphs

Abstract: A visibility drawing$$\varGamma $$ of a planar graph G maps the vertices into non-overlapping horizontal segments bars, and the edges into vertical segments visibilities, each connecting the two bars corresponding to its two end-vertices.

Swath-acquisition planning in multiple-satellite missions: an exact and heuristic approach

Abstract: This paper deals with the swath acquisition planning problem for multisatellite Earth observation missions. Given a set of satellites and a mission time frame, the problem we solve consists of selecting a set of acquisitions from the satellites in order to cover a given region of interest during the requested time frame, optimizing a certain objective function.We show that the planning problem can be modeled as a set covering problem, using basic tools of mathematical programming. The formulation of the model requires the solution of a complex computational geometry problem, and therefore the use of heuristics and metaheuristics applies. In this paper, we discuss the efficiency of the constructive phase of a greedy randomized adaptive search procedure algorithm. Computational results comparing the heuristic algorithms with the exact approach are presented.

13th CIRP Conference on Computer Aided Tolerancing Tolerance Analysis of rotating Mechanism based on Skin Model Shapes in discrete Geometry

Abstract: Geometric deviations are inevitably observable on every manufactured workpiece. These deviations affect the function and quality of mechanical products and have therefore to be controlled by geometric tolerances. Computer-aided tolerancing aims at supporting design, manufacturing, and inspection by determining and quantifying these effects of geometric deviations on the product quality and the functional behaviour. However, most established tolerance representation schemes imply abstractions of geometric deviations and are not conform with the standards for geometric dimensioning and tolerancing. These limitations led to the development of a Skin Model inspired framework for the tolerance analysis, which is based on a representation of non-ideal workpieces employing discrete geometry representation schemes, such as point clouds and surface meshes. In this contribution, this Skin Model inspired framework for computer aided tolerancing is extended to systems in motion and applied to the tolerance analysis of rotating mechanism with higher kinematic pairs. For this purpose, the generation of non-ideal part representatives, as well as their processing with algorithms for registration and computational geometry are highlighted. Finally, the results are visualized and interpreted. The procedure as well as the simulation model itself are shown in a case study of a disk cam mechanism.

Net and Prune: A Linear Time Algorithm for Euclidean Distance Problems

Abstract: We provide a general framework for getting expected linear time constant factor approximations (and in many cases FPTAS's) to several well known problems in Computational Geometry, such as k-center clustering and farthest nearest neighbor. The new approach is robust to variations in the input problem, and yet it is simple, elegant, and practical. In particular, many of these well studied problems which fit easily into our framework, either previously had no linear time approximation algorithm, or required rather involved algorithms and analysis. A short list of the problems we consider include farthest nearest neighbor, k-center clustering, smallest disk enclosing k points, kth largest distance, kth smallest m-nearest neighbor distance, kth heaviest edge in the MST and other spanning forest type problems, problems involving upward closed set systems, and more. Finally, we show how to extend our framework such that the linear running time bound holds with high probability.

Constructing Intrinsic Delaunay Triangulations from the Dual of Geodesic Voronoi Diagrams

Abstract: Intrinsic Delaunay triangulation (IDT) is a fundamental data structure in computational geometry and computer graphics. However, except for some theoretical results, such as existence and uniqueness, little progress has been made towards computing IDT on simplicial surfaces. To date the only way for constructing IDTs is the edge-flipping algorithm, which iteratively flips the non-Delaunay edge to be locally Delaunay. Although the algorithm is conceptually simple and guarantees to stop in finite steps, it has no known time complexity. Moreover, the edge-flipping algorithm may produce non-regular triangulations, which contain self-loops and/or faces with only two edges. In this paper, we propose a new method for constructing IDT on manifold triangle meshes. Based on the duality of geodesic Voronoi diagrams, our method can guarantee the resultant IDTs are regular. Our method has a theoretical worst-case time complexity $O(n^2\log n)$ for a mesh with $n$ vertices. We observe that most real-world models are far from their Delaunay triangulations, thus, the edge-flipping algorithm takes many iterations to fix the non-Delaunay edges. In contrast, our method is non-iterative and insensitive to the number of non-Delaunay edges. Empirically, it runs in linear time $O(n)$ on real-world models.

Automated Classification of Passing in Football

Abstract: A knowledgeable observer of a game of football (soccer) can make a subjective evaluation of the quality of passes made between players during the game. In this paper we consider the problem of producing an automated system to make the same evaluation of passes. We present a model that constructs numerical predictor variables from spatiotemporal match data using feature functions based on methods from computational geometry, and then learns a classification function from labelled examples of the predictor variables. In addition, we show that the predictor variables computed using methods from computational geometry are among the most important to the learned classifiers.

Tradeoffs for Space, Time, Data and Risk in Unsupervised Learning

Abstract: Faced with massive data, is it possible to trade off (statistical) risk, and (computational) space and time? This challenge lies at the heart of large-scale machine learning. Using k-means clustering as a prototypical unsupervised learning problem, we show how we can strategically summarize the data (control space) in order to trade off risk and time when data is generated by a probabilistic model. Our summarization is based on coreset constructions from computational geometry. We also develop an algorithm, TRAM, to navigate the space/time/data/risk tradeoff in practice. In particular, we show that for a fixed risk (or data size), as the data size increases (resp. risk increases) the running time of TRAM decreases. Our extensive experiments on real data sets demonstrate the existence and practical utility of such tradeoffs, not only for k-means but also for Gaussian Mixture Models.

Max-sum diversity via convex programming

Abstract: Diversity maximization is an important concept in information retrieval, computational geometry and operations research. Usually, it is a variant of the following problem: Given a ground set, constraints, and a function f that measures diversity of a subset, the task is to select a feasible subset S such that f(S) is maximized. The sum-dispersion function f(S) which is the sum of the pairwise distances in S, is in this context a prominent diversification measure. The corresponding diversity maximization is the "max-sum" or "sum-sum" diversification. Many recent results deal with the design of constant-factor approximation algorithms of diversification problems involving sum-dispersion function under a matroid constraint. In this paper, we present a PTAS for the max-sum diversity problem under a matroid constraint for distances d(.,.) of negative type. Distances of negative type are, for example, metric distances stemming from the l_2 and l_1 norms, as well as the cosine or spherical, or Jaccard distance which are popular similarity metrics in web and image search. Our algorithm is based on techniques developed in geometric algorithms like metric embeddings and convex optimization. We show that one can compute a fractional solution of the usually non-convex relaxation of the problem which yields an upper bound on the optimum integer solution. Starting from this fractional solution, we employ a deterministic rounding approach which only incurs a small loss in terms of objective, thus leading to a PTAS. This technique can be applied to other previously studied variants of the max-sum dispersion function, including combinations of diversity with linear-score maximization, improving over the previous constant-factor approximation algorithms.

A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

Abstract: A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing shows how to use a collection of mathematical techniques to solve important problems in applied mathematics and computer science areas. The book discusses fundamental tools in analytical geometry and linear algebra. It covers a wide range of topics, from matrix decomposition to curvature analysis and principal component analysis to dimensionality reduction. Written by a team of highly respected professors, the book can be used in a one-semester, intermediate-level course in computer science. It takes a practical problem-solving approach, avoiding detailed proofs and analysis. Suitable for readers without a deep academic background in mathematics, the text explains how to solve non-trivial geometric problems. It quickly gets readers up to speed on a variety of tools employed in visual computing and applied geometry.