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AN ALGORITHM TO DETERMINE THE DIRECTIONAL RELATIONSHIP BETWEEN ARBITRARILY-SHAPED POLYGONS IN THE PLANE

PATTERN RECOGNITION 1987年第1期20卷 65-74页

作者： PEUQUET, DJ ZHAN, CX UNIV CALIF SANTA BARBARA DEPT GEOGSANTA BARBARACA 93106

The directional relationship between two polygons (e.g. left, above, beside, east, north) is an important spatial property and can also be used as a selection criterion for retrieving objects from a spatial database. If the database is large, this could help significantly in speeding search by reducing the size of the necessary search space. This paper builds upon past work to develop a model for determining the directional relationship in 2-D space between two simply-connected polygons of arbitrary shape, size and distance from each other. The model is based on visual interpretation and is data structure independent. The model is also stated in algorithmic terms, and is found to have a computational complexity of O(n), where n is the total number of vertices or cells used to represent the two polygons.

关键词： Spatial relationships Direction Computer vision Geographic information systems geometric algorithms

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Efficient Geometry Compression for GPU-based Decoding in Realtime Terrain Rendering

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COMPUTER GRAPHICS FORUM 2009年第1期28卷 67-83页

作者： Dick, Christian Schneider, Jens Westermann, Ruediger Tech Univ Munich Comp Graph & Visualizat Grp D-8000 Munich Germany

We present a geometry compression scheme for restricted quadtree meshes and use this scheme for the compression of adaptively triangulated digital elevation models (DEMs). A compression factor of 8-9 is achieved by employing a generalized strip representation of quadtree meshes to incrementally encode vertex positions. In combination with adaptive error-controlled triangulation, this allows us to significantly reduce bandwidth requirements in the rendering of large DEMs that have to be paged from disk. The compression scheme is specifically tailored for GPU-based decoding, since it minimizes dependent memory access operations. We can thus trade CPU operations and CPU-GPU data transfer for GPU processing, resulting in twice faster streaming of DEMs from main memory into GPU memory. A novel storage format for decoded DEMs on the GPU facilitates a sustained rendering throughput of about 300 million triangles per second. Due to these properties, the proposed scheme enables scalable rendering with respect to the display resolution independent of the data size. For a maximum screen-space error below 1 pixel it achieves frame rates of over 100 fps, even on high-resolution displays. We validate the efficiency of the proposed method by presenting experimental results on scanned elevation models of several hundred gigabytes.

关键词： terrain rendering geometry compression GPU methods restricted quadtree meshes level-of-detail control out-of-core rendering E 4 [Data]: Coding and Information Theory - Data Compression I 3 3 [Computer Graphics]: Picture Image Generation - Viewing algorithms I 3 5 [Computer Graphics]: Computational Geometry and Object Modeling - Surface Representations geometric algorithms Object Hierarchies I 3 7 [Computer Graphics]: Three-Dimensional Graphics and Realism - Virtual Reality

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Exact regional visibility using line space partitioning

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COMPUTERS & GRAPHICS-UK 2003年第4期27卷 569-580页

作者： Bittner, J Prikryl, J Slavík, P Vienna Univ Technol Inst Comp Graph & Algorithms A-1040 Vienna Austria Czech Tech Univ Ctr Appl Cybernet Prague 12135 2 Czech Republic Czech Tech Univ Dept Appl Math Prague 11000 1 Czech Republic Czech Tech Univ Dept Comp Sci Prague 12135 2 Czech Republic

We present a new method for computing visibility from a polygonal region in the plane considering a set of line segments as occluders. The proposed method provides a comprehensive description of visibility from the given region. We represent sets of occluded rays using a hierarchical partitioning of dual space (line space). The line space partitioning is maintained by a BSP tree that provides efficient operations on the sets of lines. The implementation shows that the method is suitable for computing potentially visible sets in large scenes with various visibility characteristics. (C) 2003 Elsevier Ltd. All rights reserved.

关键词： visible surface algorithms geometric algorithms spatial hierarchies

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Median Trajectories

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ALGORITHMICA 2013年第3期66卷 595-614页

作者： Buchin, Kevin Buchin, Maike van Kreveld, Marc Loffler, Maarten Silveira, Rodrigo I. Wenk, Carola Wiratma, Lionov TU Eindhoven Eindhoven Netherlands Univ Utrecht Utrecht Netherlands Univ Politecn Cataluna Barcelona Spain Univ Texas San Antonio San Antonio TX USA Parahyangan Catholic Univ Bandung Indonesia

We investigate the concept of a median among a set of trajectories. We establish criteria that a "median trajectory" should meet, and present two different methods to construct a median for a set of input trajectories. The first method is very simple, while the second method is more complicated and uses homotopy with respect to sufficiently large faces in the arrangement formed by the trajectories. We give algorithms for both methods, analyze the worst-case running time, and show that under certain assumptions both methods can be implemented efficiently. We empirically compare the output of both methods on randomly generated trajectories, and evaluate whether the two methods yield medians that are according to our intuition. Our results suggest that the second method, using homotopy, performs considerably better.

关键词： Trajectories geometric algorithms Homotopy

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Simplification of Articulated Meshes

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COMPUTER GRAPHICS FORUM 2009年第2期28卷 347-353页

作者： Landreneau, Eric Schaefer, Scott Texas A&M Univ College Stn TX 77843 USA

We present a method for simplifying a polygonal character with an associated skeletal deformation such that the simplified character approximates the original shape well when deformed. As input, we require a set of example poses that are representative of the types of deformations the character undergoes and we produce a multi-resolution hierarchy for the simplified character where all simplified vertices also have associated skin weights. We create this hierarchy by minimizing an error metric for a simplified set of vertices and their skin weights, and we show that this quartic error metric can be effectively minimized using alternating quadratic minimization for the vertices and weights separately. To enable efficient GPU accelerated deformations of the simplified character, we also provide a method that guarantees the maximum number of bone weights per simplified vertex is less than a user specified threshold at all levels of the hierarchy.

关键词： Computer Graphics [I.3.5]: Computational Geometry and Object Modeling geometric algorithms

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Freeform Honeycomb Structures

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COMPUTER GRAPHICS FORUM 2014年第5期33卷 185-194页

作者： Jiang, Caigui Wang, Jun Wallner, Johannes Pottmann, Helmut KAUST Thuwal Saudi Arabia Graz Univ Technol Graz Austria TU Wien Vienna Austria

Motivated by requirements of freeform architecture, and inspired by the geometry of hexagonal combs in beehives, this paper addresses torsion-free structures aligned with hexagonal meshes. Since repetitive geometry is a very important contribution to the reduction of production costs, we study in detail "honeycomb structures", which are defined as torsion-free structures where the walls of cells meet at 120 degrees. Interestingly, the Gauss-Bonnet theorem is useful in deriving information on the global distribution of node axes in such honeycombs. This paper discusses the computation and modeling of honeycomb structures as well as applications, e. g. for shading systems, or for quad meshing. We consider this paper as a contribution to the wider topic of freeform patterns, polyhedral or otherwise. Such patterns require new approaches on the technical level, e. g. in the treatment of smoothness, but they also extend our view of what constitutes aesthetic freeform geometry.

关键词： Categories and Subject Descriptors (according to ACM CCS) I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling&mdash geometric algorithms

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algorithms for selecting cutters in multi-part milling problems

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COMPUTER-AIDED DESIGN 2003年第9期35卷 825-839页

作者： Yao, ZY Gupta, SK Nau, DS Univ Maryland Dept Mech Engn Syst Res Inst College Pk MD 20742 USA Univ Maryland Dept Comp Sci Syst Res Inst College Pk MD 20742 USA

This paper describes geometric algorithms for automatically selecting an optimal sequence of cutters for machining a set of 2.5-D parts. In milling operations, cutter size affects the machining time significantly. Meanwhile, if the batch size is small, it is also important to shorten the time spent on loading tools into the tool magazine and establishing z-length compensation values. Therefore, in small-batch manufacturing, if we can select a set of milling tools that will produce good machining time on more than one type of parts, then several unnecessary machine-tool reconfiguration operations can be eliminated. In selecting milling cutters we consider both the tool loading time and the machining time and generate solutions that allow us to minimize the total machining time. In this paper we first present algorithms for finding the area that can be cut by a given cutter. Then we describe a graph search formulation for the tool selection problem. Finally, the optimal sequence of cutters is selected by using Dijkstra's shortest path planning algorithm. (C) 2003 Elsevier Science Ltd. All rights reserved.

关键词： geometric algorithms 2.5-D milling cutter selection

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MOTION PLANNING WITH INERTIAL CONSTRAINTS

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ALGORITHMICA 1987年第4期2卷 431-475页

作者： ODUNLAING, C UNIV DUBLIN TRINITY COLL DEPT MATHDUBLIN 2IRELAND

A bodyB must move from a placementZ 0 to a placementZ 1, while avoiding collision with a setS of moving obstacles. The motion must satisfy an inertial constraint: the acceleration cannot exceed a given boundM. The pro... 详细信息

A bodyB must move from a placementZ ₀ to a placementZ ₁, while avoiding collision with a setS of moving obstacles. The motion must satisfy an inertial constraint: the acceleration cannot exceed a given boundM. The problem is analyzed, and polynomial-time motion-planning algorithms are given for the case of a particle moving in one dimension.

关键词： Motion planning Robotics geometric algorithms

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Harmonic Functions for Rotational Symmetry Vector Fields

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COMPUTER GRAPHICS FORUM 2016年第7期35卷 507-516页

作者： Shen, Zhongwei Fang, Xianzhong Liu, Xinguo Bao, Hujun Huang, Jin Zhejiang Univ State Key Lab CAD&CG Hangzhou Zhejiang Peoples R China

Representing rotational symmetry vector as a set of vectors is not suitable for design due to lacking of a consistent ordering for measurement. In this paper we introduce a spectral method to find rotation invariant harmonic functions for symmetry vector field design. This method is developed for 3D vector fields, but it is applicable in 2D. Given the finite symmetry group G of a symmetry vector field v(x) on a 3D domain Omega, we formulate the harmonic function h (s) as a stationary point of group G. Using the real spherical harmonic (SH) bases, we showed the coefficients of the harmonic functions are an eigenvector of the SH rotation matrices corresponding to group G. Instead of solving eigen problems to obtain the eigenvector, we developed a forward constructive method based on orthogonal group theory. The harmonic function found by our method is not only invariant under G, but also expressive and can distinguish different rotations with respect to G. At last, we demonstrate some vector field design results with tetrahedron-symmetry, cube-symmetry and dodecahedron-symmetry groups.

关键词： Categories and Subject Descriptors (according to ACM CCS) I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling&mdash geometric algorithms

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C¹- and C²-continuous spline-interpolation of a regular triangular net of points

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COMPUTERS & GRAPHICS-UK 2003年第6期27卷 917-930页

作者： Wiltsche, A Graz Univ Technol Inst Geometry A-8010 Graz Austria

The investigations focus on the construction of a C-k-continuous (k = 0, 1, 2) interpolating spline-surface for a given data set consisting of points P-ijk, arranged in a regular triangular net and corresponding barycentric parameter triples (u(i), v(j), w(k)). We try to generalize an algorithm by A.W. Overhauser who solved the analogous problem for the case of a univariate data set. As a straightforward generalization does not work out we adapt the Overhauser-construction. We use some blending of basic surfaces with uniquely determined basic functions. This yields a spline-surface with a polynomial parametric representation which display C-1 - or C-2-continuity along the common curve of two adjacent subpatches. Local control of the emerging spline surface is provided which means moving one data point P changes only some of the sub-patches around P and does not affect regions lying far away. (C) 2003 Elsevier Ltd. All rights reserved.

关键词： spline interpolation regular triangular net geometric algorithms surface representations splines

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