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Computational geometry
About: Computational geometry is a research topic. Over the lifetime, 5110 publications have been published within this topic receiving 220642 citations.
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TL;DR: Algorithms that count the number of pairwise intersections among a set of N objects in the plane and algorithms that report all such intersections are given.
Abstract: An interesting class of "geometric intersection problems" calls for dealing with the pairwise intersections among a set of N objects in the plane, These problems arise in many applications such as printed circuit design, architectural data bases, and computer graphics. Shamos and Hoey have described a number of algorithms for detecting whether any two objects in a planar set intersect. In this paper we extend their work by giving algorithms that count the number of such intersections and algorithms that report all such intersections.
1,062 citations
17 Jul 2013
TL;DR: The Multi-Parametric Toolbox is a collection of algorithms for modeling, control, analysis, and deployment of constrained optimal controllers developed under Matlab that features a powerful geometric library that extends the application of the toolbox beyond optimal control to various problems arising in computational geometry.
Abstract: The Multi-Parametric Toolbox is a collection of algorithms for modeling, control, analysis, and deployment of constrained optimal controllers developed under Matlab. It features a powerful geometric library that extends the application of the toolbox beyond optimal control to various problems arising in computational geometry. The new version 3.0 is a complete rewrite of the original toolbox with a more flexible structure that offers faster integration of new algorithms. The numerical side of the toolbox has been improved by adding interfaces to state of the art solvers and by incorporation of a new parametric solver that relies on solving linear-complementarity problems. The toolbox provides algorithms for design and implementation of real-time model predictive controllers that have been extensively tested.
1,054 citations
03 Aug 1997
TL;DR: This paper defines efficient refinement criteria based on the view frustum, surface orientation, and screen-space geometric error, and develops a real-time algorithm for incrementally refining and coarsening the mesh according to these criteria.
Abstract: Level-of-detail (LOD) representations are an important tool for realtime rendering of complex geometric environments. The previously introduced progressive mesh representation defines for an arbitrary triangle mesh a sequence of approximating meshes optimized for view-independent LOD. In this paper, we introduce a framework for selectively refining an arbitrary progressive mesh according to changing view parameters. We define efficient refinement criteria based on the view frustum, surface orientation, and screen-space geometric error, and develop a real-time algorithm for incrementally refining and coarsening the mesh according to these criteria. The algorithm exploits view coherence, supports frame rate regulation, and is found to require less than 15% of total frame time on a graphics workstation. Moreover, for continuous motions this work can be amortized over consecutive frames. In addition, smooth visual transitions (geomorphs) can be constructed between any two selectively refined meshes. A number of previous schemes create view-dependent LOD meshes for height fields (e.g. terrains) and parametric surfaces (e.g. NURBS). Our framework also performs well for these special cases. Notably, the absence of a rigid subdivision structure allows more accurate approximations than with existing schemes. We include results for these cases as well as for general meshes. CR Categories: I.3.3 [Computer Graphics]: Picture/Image Generation Display algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling surfaces and object representations. Additional
973 citations
TL;DR: This work develops and analyzes a method, based on bounding-volume hierarchies, for efficient collision detection for objects moving within highly complex environments, and provides experimental evidence showing that this approach yields substantially faster collision detection than previous methods.
Abstract: Collision detection is of paramount importance for many applications in computer graphics and visualization. Typically, the input to a collision detection algorithm is a large number of geometric objects comprising an environment, together with a set of objects moving within the environment. In addition to determining accurately the contacts that occur between pairs of objects, one needs also to do so at real-time rates. Applications such as haptic force feedback can require over 1000 collision queries per second. We develop and analyze a method, based on bounding-volume hierarchies, for efficient collision detection for objects moving within highly complex environments. Our choice of bounding volume is to use a discrete orientation polytope (k-DOP), a convex polytope whose facets are determined by halfspaces whose outward normals come from a small fixed set of k orientations. We compare a variety of methods for constructing hierarchies (BV-trees) of bounding k-DOPs. Further, we propose algorithms for maintaining an effective BV-tree of k-DOPs for moving objects, as they rotate, and for performing fast collision detection using BV-trees of the moving objects and of the environment. Our algorithms have been implemented and tested. We provide experimental evidence showing that our approach yields substantially faster collision detection than previous methods.
941 citations
TL;DR: It is suggested that B6zier curves should be considered as a single type of curve, but the design of these curves should not be confused with other types of curves, which are known to have different properties.
839 citations