Computational geometry.
01 Jan 1978-
About: The article was published on 1978-01-01 and is currently open access. It has received 366 citations till now. The article focuses on the topics: Computational geometry.
Citations
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15 Apr 1995
TL;DR: An exact and interactive collision detection system for large-scale environments, I-COLLIDE, based on pruning multiple-object pairs using bounding boxes and performing exact collision detection between selected pairs of polyhedral models.
Abstract: we present an exact and interactive collision detection system, I-COLLIDE, for large-scale environments. Such environments are characterized by the number of objects undergoing rigid motion and the complexity of the models. The algorithm does not assume the objects' motions can be expressed as a closed form function of time. The collision detection system is general and can be easily interfaced with a variety of applications. The algorithm uses a two-level approach based on pruning multiple-object pairs using bounding boxes and performing exact collision detection between selected pairs of polyhedral models. We demonstrate the performance of the system in walkthrough and simulation environments consisting of a large number of moving objects. In particular, the system takes less than 1/20 of a second to determine all the collisions and contacts in an environment consisting of more than 1000 moving polytopes, each consisting of more than 50 faces on an HP-9000/750.
753 citations
Cites background from "Computational geometry."
...The interval tree is a common data structure for performing such two-dimensional range queries [22]....
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...At the same time, the emphasis in the computational geometry has been on theoretically e cient intersection detection algorithms [22]....
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TL;DR: These apprenticeship learning algorithms have enabled us to significantly extend the state of the art in autonomous helicopter aerobatics, including the first autonomous execution of a wide range of maneuvers, including in-place flips, in- place rolls, loops and hurricanes.
Abstract: Autonomous helicopter flight is widely regarded to be a highly challenging control problem. Despite this fact, human experts can reliably fly helicopters through a wide range of maneuvers, including aerobatic maneuvers at the edge of the helicopterâs capabilities. We present apprenticeship learning algorithms, which leverage expert demonstrations to efficiently learn good controllers for tasks being demonstrated by an expert. These apprenticeship learning algorithms have enabled us to significantly extend the state of the art in autonomous helicopter aerobatics. Our experimental results include the first autonomous execution of a wide range of maneuvers, including but not limited to in-place flips, in-place rolls, loops and hurricanes, and even auto-rotation landings, chaos and tic-tocs, which only exceptional human pilots can perform. Our results also include complete airshows, which require autonomous transitions between many of these maneuvers. Our controllers perform as well as, and often even better than, our expert pilot.
630 citations
TL;DR: Given a triangulation of a simple polygonP, linear-time algorithms for solving a collection of problems concerning shortest paths and visibility withinP are presented.
Abstract: Given a triangulation of a simple polygonP, we present linear-time algorithms for solving a collection of problems concerning shortest paths and visibility withinP. These problems include calculation of the collection of all shortest paths insideP from a given source vertexS to all the other vertices ofP, calculation of the subpolygon ofP consisting of points that are visible from a given segment withinP, preprocessingP for fast "ray shooting" queries, and several related problems.
544 citations
Cites background from "Computational geometry."
...115 of [25]); thus each (unsuccessful) "comparison" performed at some node v* during this search determines a unique side of v* in which the desired v lies, so that binary search is applicable....
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TL;DR: Robust and time-efficient skeletonization of a (planar) shape can be achieved by first regularizing the Voronoi diagram of a shape's boundary points and then by establishing a hierarchic organization of skeleton constituents.
422 citations
Cites background or methods from "Computational geometry."
...Data structures such as the Voronoi diagram or the Voronoi skeleton are described by means of attributed doubly-connected-edge-lists (DCEL).(58) The DCEL of a Voronoi diagram consists of three separate lists DCEL(Vor) = (P;V; E)....
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...The Voronoi diagram (VD) is a well-known tool in Computational Geometry (for details, see(58) or,(63) for VD construction algorithms(58, 64–67))....
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...The solution to the above problem results in a net-shaped partition of the plane into Voronoi polygons,(58) polygonal regions bounded by straight lines, rays or straight line segments....
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01 Jan 1993
TL;DR: An opportunistic global path planner algorithm which uses the incremental distance computation algorithm to trace out a one-dimensional skeleton for the purpose of robot motion planning and its performance attests their promise for real-time dynamic simulations as well as applications in a computer generated virtual environment.
Abstract: We present efficient algorithms for collision detection and contact determination between geometric models, described by linear or curved boundaries, undergoing rigid motion. The heart of our collision detection algorithm is a simple and fast incremental method to compute the distance between two convex polyhedra. It utilizes convexity to establish some local applicability criteria for verifying the closest features. A preprocessing procedure is used to subdivide each feature's neighboring features to a constant size and thus guarantee expected constant running time for each test.
The expected constant time performance is an attribute from exploiting the geometric coherence and locality. Let n be the total number of features, the expected run time is between $O(\sqrt{n})$ and O(n) depending on the shape, if no special initialization is done. This technique can be used for dynamic collision detection, planning in three-dimensional space, physical simulation, and other robotics problems.
The set of models we consider includes polyhedra and objects with surfaces described by rational spline patches or piecewise algebraic functions. We use the expected constant time distance computation algorithm for collision detection between convex polyhedral objects and extend it using a hierarchical representation to distance measurement between non-convex polytopes. Next, we use global algebraic methods for solving polynomial equations and the hierarchical description to devise efficient algorithms for arbitrary curved objects.
We also describe two different approaches to reduce the frequency of collision detection from$$N\choose2$$pairwise comparisons in an environment with n moving objects. One of them is to use a priority queue sorted by a lower bound on time to collision; the other uses an overlap test on bounding boxes.
Finally, we present an opportunistic global path planner algorithm which uses the incremental distance computation algorithm to trace out a one-dimensional skeleton for the purpose of robot motion planning.
The performance of the distance computation and collision detection algorithms attests their promise for real-time dynamic simulations as well as applications in a computer generated virtual environment.
290 citations
Cites background or methods from "Computational geometry."
...But, if they are parallel yet not overlapping, then we use a linear time procedure [71] to nd the closest pair of edges, say EA and EB between FA and FB, then invoke the algorithm again with this new pair of candidates (EA; EB)....
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...[71] F. P. Preparata and M. I. Shamos....
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...Some of the materials presented in this chapter can be found in the books byHo mann, Preparata and Shamos [46, 71]....
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...38 An elegant approach which runs in O(M + N) can be found in [66, 71], where the two polygons A and B each has M , N vertices respectively....
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...Please refer to [66, 71] for all the details....
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References
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TL;DR: In this paper, the basic problem of interconnecting a given set of terminals with a shortest possible network of direct links is considered, and a set of simple and practical procedures are given for solving this problem both graphically and computationally.
Abstract: The basic problem considered is that of interconnecting a given set of terminals with a shortest possible network of direct links Simple and practical procedures are given for solving this problem both graphically and computationally It develops that these procedures also provide solutions for a much broader class of problems, containing other examples of practical interest
4,395 citations
TL;DR: P can be chosen to I&E the centroid oC the triangle formed by X, y and z and Express each si E S in polar coordinates th origin P and 8 = 0 in the direction of zu~ arhitnry fixed half-line L from P.
1,741 citations
TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A, B, C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than ${2n/3}$ vertices, and C contains no more than $2.
Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.
1,312 citations
01 Oct 1977
TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A,B,C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than $2.
Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A,B,C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than $2\sqrt{2}\sqrt{2}$ vertices. We exhibit an algorithm which finds such a partition A,B,C in O(n) time.
1,264 citations
TL;DR: Any n-vertex planar graph has the property that it can be divided into components of roughly equal size by removing only O(√n) vertices, and this separator theorem in combination with a divide-and-conquer strategy leads to many new complexity results for planar graphs problems.
Abstract: Any n-vertex planar graph has the property that it can be divided into components of roughly equal size by removing only $O(\sqrt n )$ vertices. This separator theorem, in combination with a divide-and-conquer strategy, leads to many new complexity results for planar graph problems. This paper describes some of these results.
767 citations