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Journal Article

Triangulating a simple polygon in linear time

01 Jan 1991-IEEE Transactions on Industry Applications (Institute of Electrical and Electronics Engineers Inc.)-Vol. 27, pp 220-230
TL;DR: A deterministic algorithm for triangulating a simple polygon in linear time is given, using the polygon-cutting theorem and the planar separator theorem, whose role is essential in the discovery of new diagonals.
Abstract: We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.
Citations
More filters
MonographDOI
01 Jan 2006
TL;DR: This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms, into planning under differential constraints that arise when automating the motions of virtually any mechanical system.
Abstract: Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

6,340 citations


Cites background from "Triangulating a simple polygon in l..."

  • ...For example, one of the most famous algorithms from computational geometry can split a simple2 polygon into triangles in O(n) time for a polygon with n edges [192]....

    [...]

  • ...If Cfree is simply connected, then, surprisingly, a triangulation can be computed in linear time [192]....

    [...]

Book
10 Oct 2002
TL;DR: Geometric Tools for Computer Graphics is an extensive, conveniently organized collection of proven solutions to fundamental problems that you'd rather not solve over and over again, including building primitives, distance calculation, approximation, containment, decomposition, intersection determination, separation, and more.
Abstract: From the Publisher: Do you spend too much time creating the building blocks of your graphics applications or finding and correcting errors? Geometric Tools for Computer Graphics is an extensive, conveniently organized collection of proven solutions to fundamental problems that you'd rather not solve over and over again, including building primitives, distance calculation, approximation, containment, decomposition, intersection determination, separation, and more. If you have a mathematics degree, this book will save you time and trouble. If you don't, it will help you achieve things you may feel are out of your reach. Inside, each problem is clearly stated and diagrammed, and the fully detailed solutions are presented in easy-to-understand pseudocode. You also get the mathematics and geometry background needed to make optimal use of the solutions, as well as an abundance of reference material contained in a series of appendices. KEY FEATURES: * Filled with robust, thoroughly tested solutions that will save you time and help you avoid costly errors. * Covers problems relevant for both 2D and 3D graphics programming. * Presents each problem and solution in stand-alone form allowing you the option of reading only those entries that matter to you. * Provides the math and geometry background you need to understand the solutions and put them to work. * Clearly diagrams each problem and presents solutions in easy-to-understand pseudocode. * Resources associated with the book are available at the companion Web site. Author Biography: Philip Schneider leads a modeling and dynamic simulation software group at Walt Disney Feature Animation. Prior to that, his work at Apple and Digital Equipment Corporation in 3D graphics ranged from low-level interfaces to graphics libraries and interactive applications. He holds an M.S. in Computer Science from the University of Washington. Dave Eberly is the president of Magic Software, Inc., a company known for its free source code and documentation for computer graphics, image analysis, and numerical methods. Previously, he was the director of engineering at Numerical Design Limited, the company responsible for the real-time 3D game engine, NetImmerse. His background includes a B.A. in mathematics from Bloomsburg University, M.S. and Ph.D. degrees in mathematics from the University of Colorado at Boulder, and M.S. and Ph.D. degrees in computer science from the University of North Carolina at Chapel Hill. Dave is author of 3D Game Engine Design, co-author with Philip Schneider of Geometric Tools for Computer Graphics, and author of the forthcoming Game Physics (Spring 2003).

597 citations

Proceedings ArticleDOI
01 Jul 2000
TL;DR: An object-space morphing technique that blends the interiors of given two- or three-dimensional shapes rather than their boundaries that is rigid in the sense that local volumes are least-distorting as they vary from their source to target configurations is presented.
Abstract: We present an object-space morphing technique that blends the interiors of given two- or three-dimensional shapes rather than their boundaries. The morph is rigid in the sense that local volumes are least-distorting as they vary from their source to target configurations. Given a boundary vertex correspondence, the source and target shapes are decomposed into isomorphic simplicial complexes. For the simplicial complexes, we find a closed-form expression allocating the paths of both boundary and interior vertices from source to target locations as a function of time. Key points are the identification of the optimal simplex morphing and the appropriate definition of an error functional whose minimization defines the paths of the vertices. Each pair of corresponding simplices defines an affine transformation, which is factored into a rotation and a stretching transformation. These local transformations are naturally interpolated over time and serve as the basis for composing a global coherent least-distorting transformation.

554 citations


Cites background from "Triangulating a simple polygon in l..."

  • ...Triangulating a single polygon is possible using only the vertices of the polygon (e.g. [5])....

    [...]

Book ChapterDOI
01 Jan 2000
TL;DR: Since Victor Klee's question, numerous variations on the art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others.
Abstract: In 1973, Victor Klee posed the following question: How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naive question of combinatorial geometry has, since its formulation, stimulated a plethora of papers, surveys and a book, most of them written in the last fifteen years. The first result in this area, due to V. Chvatal, asserts that n 3 guards are occasionally necessary and always sufficient to guard an art gallery represented by a simple polygon with n vertices. Since ChvataFs result, numerous variations on the art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this paper, we survey most of these results.

474 citations

Journal ArticleDOI
TL;DR: A large class of problems is described for which it is proved that they are all at least as difficult as the following base problem 3sum: Given a set S of n integers, are there three elements of S that sum up to 0.
Abstract: There are many problems in computational geometry for which the best know algorithms take time @Q(n^2) (or more) in the worst case while only very low lower bounds are known. In this paper we describe a large class of problems for which we prove that they are all at least as difficult as the following base problem 3sum: Given a set S of n integers, are there three elements of S that sum up to 0. We call such problems 3sum-hard. The best known algorithm for the base problem takes @Q(n^2) time. The class of 3sum-hard problems includes problems like: Given a set of lines in the plane, are there three that meet in a point?; or: Given a set of triangles in the plane, does their union have a hole? Also certain visibility and motion planning problems are shown to be in the class. Although this does not prove a lower bound for these problems, there is no hope of obtaining o(n^2) solutions for them unless we can improve the solution for the base problem.

429 citations

References
More filters
MonographDOI
01 Jan 2006
TL;DR: This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms, into planning under differential constraints that arise when automating the motions of virtually any mechanical system.
Abstract: Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

6,340 citations

Book
01 Jan 1987
TL;DR: In this paper, the authors proposed a visibility algorithm based on three-dimensions and miscellany of the polygons, and showed that minimal guard covers threedimensions of the polygon.
Abstract: Polygon partitions Orthogonal polygons Mobile guards Miscellaneous shapes Holes Exterior visibility Visibility groups Visibility algorithms Minimal guard covers Three-dimensions and miscellany.

1,547 citations

Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: This work presents a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage.
Abstract: A planar subdivision is any partition of the plane into (possibly unbounded) polygonal regions. The subdivision search problem is the following: given a subdivision $S$ with $n$ line segments and a query point $p$, determine which region of $S$ contains $p$. We present a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage. Our subdivision search structure can be constructed in linear time from the subdivision representation used in many applications.

810 citations