Topic
Minkowski addition
About: Minkowski addition is a research topic. Over the lifetime, 789 publications have been published within this topic receiving 16021 citations.
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Book•
01 Feb 1993TL;DR: Inequalities for mixed volumes 7. Selected applications Appendix as discussed by the authors ] is a survey of mixed volumes with bounding boxes and quermass integrals, as well as a discussion of their applications.
Abstract: 1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.
3,954 citations
09 Mar 2005
TL;DR: The main innovation of the method consists in the use of zonotopes for reachable set representation, which has been used to treat several examples and has shown great performances for high dimensional systems.
Abstract: We present a method for the computation of reachable sets of uncertain linear systems. The main innovation of the method consists in the use of zonotopes for reachable set representation. Zonotopes are special polytopes with several interesting properties : they can be encoded efficiently, they are closed under linear transformations and Minkowski sum. The resulting method has been used to treat several examples and has shown great performances for high dimensional systems. An extension of the method for the verification of piecewise linear hybrid systems is proposed.
614 citations
TL;DR: The second part of a series of surveys on the geometry of finite dimensional Banach spaces (Minkowski spaces) is presented in this article, where the authors discuss results that refer to the following three topics: bodies of constant Minkowski width, generalized convexity notions that are important for Minkowowski spaces, and bisectors as well as Voronoi diagrams in Minkowsky spaces.
305 citations
24 Apr 2000
TL;DR: These algorithms have been used to perform proximity queries for applications including virtual prototyping, dynamic simulation, and motion planning on complex models and have achieved significant speedups on many benchmarks.
Abstract: We present new distance computation algorithms using hierarchies of rectangular swept spheres. Each bounding volume of the tree is described as the Minkowski sum of a rectangle and a sphere, and fits tightly to the underlying geometry. We present accurate and efficient algorithms to build the hierarchies and perform distance queries between the bounding volumes. We also present traversal techniques for accelerating distance queries using coherence and priority directed search. These algorithms have been used to perform proximity queries for applications including virtual prototyping, dynamic simulation, and motion planning on complex models. As compared to earlier algorithms based on bounding volume hierarchies for separation distance and approximate distance computation, our algorithms have achieved significant speedups on many benchmarks.
303 citations
Book•
27 Oct 2003
TL;DR: This chapter discusses the design and implementation of the GJK Algorithm, which automates the very labor-intensive and therefore time-heavy and expensive process of manually partitioning polytopes into discrete components.
Abstract: 1 Introduction 1.1 Problem Domain 1.2 Historical Background 1.3 Organization 2 Concepts 2.1 Geometry 2.1.1 Notational Conventions 2.1.2 Vector Spaces 2.1.3 Affine Spaces 2.1.4 Euclidean Spaces 2.1.5 Affine Transformations 2.1.6 Three-dimensional Space 2.2 Objects 2.2.1 Polytopes 2.2.2 Polygons 2.2.3 Quadrics 2.2.4 Minkowski Addition 2.2.5 Complex Shapes and Scenes 2.3 Animation 2.4 Time 2.5 Response 2.6 Performance 2.6.1 Frame Coherence 2.6.2 Geometric Coherence 2.6.3 Average Time 2.7 Robustness 2.7.1 Floating-Point Numbers 2.7.2 Stability 2.7.3 Coping with Numerical Problems 3 Basic Primitives 3.1 Spheres 3.1.1 Sphere-Sphere Test 3.1.2 Ray-Sphere Test 3.1.3 Line-Segment-Sphere Test 3.2 Axis-Aligned Boxes 3.2.1 Ray-Box Test 3.2.2 Sphere-Box Test 3.3 Separating Axes 3.3.1 Line-Segment-Box Test 3.3.2 Triangle-Box Test 3.3.3 Box-Box Test 3.4 Polygons 3.4.1 Ray-Triangle Test 3.4.2 Line Segment-Triangle Test 3.4.3 Ray-Polygon Test 3.4.4 Triangle-Triangle Test 3.4.5 Polygon-Polygon Test 3.4.6 Triangle-Sphere Test 3.4.7 Polygon-Volume Tests 4 Convex Objects 4.1 Proximity Queries 4.2 Overview of Algorithms for Polytopes 4.2.1 Finding a Common Point 4.2.2 Finding a Separating Plane 4.2.3 Distance and Penetration Depth Computation 4.3 The Gilbert-Johnson-Keerthi Algorithm 4.3.1 Overview 4.3.2 Convergence and Termination 4.3.3 Johnson's Distance Algorithm 4.3.4 Support Mappings 4.3.5 Implementing the GJK Algorithm 4.3.6 Numerical Aspects of the GJK Algorithm 4.3.7 Testing for Intersections 4.3.8 Penetration Depth 5 Spatial Data Structures 5.1 Nonconvex Polyhedra 5.1.1 Convex Decomposition 5.1.2 Polyhedral Surfaces 5.1.3 Point in Nonconvex Polyhedron 5.2 Space Partitioning 5.2.1 Voxel Grids 5.2.2 Octrees and k-d Trees 5.2.3 Binary Space Partitioning Trees 5.2.4 Discussion 5.3 Model Partitioning 5.3.1 Bounding Volumes 5.3.2 Bounding-Volume Hierarchies 5.3.3 AABB Trees versus OBB Trees 5.3.4 AABB Trees and Deformable Models 5.4 Broad Phase 5.4.1 Sweep and Prune 5.4.2 Implementing the Sweep-and-Prune Algorithm 5.4.3 Ray Casting and AABBs 6 Design of SOLID 6.1 Requirements 6.2 Overview of SOLID 6.3 Design Decisions 6.3.1 Shape Representation 6.3.2 Motion Specification 6.3.3 Response Handling 6.3.4 Algorithms 6.4 Evaluation 6.5 Implementation Notes 6.5.1 Generic Data Types and Algorithms 6.5.2 Fundamental 3D Classes 7 Conclusion 7.1 State of the Art 7.2 Future Work Bibliography Index About the CD-ROM Trademarks
301 citations