An optimal visibility graph algorithm for triangulated simple polygons

LetP be a triangulated simple polygon withn sides. The visibility graph ofP has an edge between every pair of polygon vertices that can be connected by an open segment in the interior ofP. We describe an algorithm that finds the visibility graph ofP inO(m) time, wherem is the number of edges in the visibility graph. Becausem can be as small asO(n), the algorithm improves on the more general visibility algorithms of Asanoet al. [AAGHI] and Welzl [W], which take (n ²) time, and on Suri'sO(m logn) visibility graph algorithm for simple polygons [S].This work was supported in part by a U.S. Army Research Office fellowship under agreement DAAG29-83-G-0020.     

Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons

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.Work on this paper by this author has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, the IBM Corporation, and from the U.S.-Israel Binational Science Foundation.Work on this paper by this author has been supported by National Science Foundation Grant DCR-86-05962.     

Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons

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.     

A note on the combinatorial structure of the visibility graph in simple polygons

Combinatorial structure of visibility is probably one of the most fascinating and interesting areas of engineering and computer science. The usefulness of visibility graphs in computational geometry and robotic navigation problems like motion planning, unknown-terrain learning, shortest-path planning, etc., cannot be overstressed. The visibility graph, apart from being an important data structure for storing and updating geometric information, is a valuable mathematical tool in probing and understanding the nature of shapes of polygonal and polyhedral objects. In this research we wish to initially focus our attention on a fundamental class of geometric objects. These geometric objects may be looked upon as building blocks for more complex geometric objects, and which offer an ideal balance between complexity and simplicity, namely simple polygons.

A major theme of the proposed paper is the investigation of the combinatorial structure of the visibility graph. More importantly, the goals of this paper are:

1. (i) To characterize the visibility graphs of simple polygons by obtaining necessary and sufficient conditions a graph must satisfy to qualify for the visibility graph of a simple polygon

2. (ii) To obtain hierarchical relationships between visibility graphs of simple polygons of a given number of vertices by treating them as representing simple polygons that are deformations of one another.

3. (iii) To exploit the potential of complete graphs to be natural coordinate systems for addressing the problem of reconstructing a simple polygon from visibility graph.

We intend to achieve this by defining appropriate “betweenness” relationships on points with respect to the edges of the complete graphs.     

Parallel methods for visibility and shortest-path problems in simple polygons

In this paper we give efficient parallel algorithms for solving a number of visibility and shortest-path problems for simple polygons. Our algorithms all run inO(logn) time and are based on the use of a new data structure for implicitly representing all shortest paths in a simple polygonP, which we call thestratified decomposition tree. We use this approach to derive efficient parallel methods for computing the visibility ofP from an edge, constructing the visibility graph of the vertices ofP (using an output-sensitive number of processors), constructing the shortest-path tree from a vertex ofP, and determining all-farthest neighbors for the vertices inP. The computational model we use is the CREW PRAM.     

Parallel methods for visibility and shortest-path problems in simple polygons

In this paper we give efficient parallel algorithms for solving a number of visibility and shortest-path problems for simple polygons. Our algorithms all run inO(logn) time and are based on the use of a new data structure for implicitly representing all shortest paths in a simple polygonP, which we call thestratified decomposition tree. We use this approach to derive efficient parallel methods for computing the visibility ofP from an edge, constructing the visibility graph of the vertices ofP (using an output-sensitive number of processors), constructing the shortest-path tree from a vertex ofP, and determining all-farthest neighbors for the vertices inP. The computational model we use is the CREW PRAM.This research was announced in preliminary form in theProceedings of the 6th ACM Symposium on Computational Geometry, 1990, pp. 73–82. The research of Michael T. Goodrich was supported by the National Science Foundation under Grants CCR-8810568 and CCR-9003299, and by the NSF and DARPA under Grant CCR-8908092.     

An Optimal Algorithm for Solving Collision Distance Between Convex Polygons in Plane

In this paper,we study the problem of calculating th minimum collision distance between two planar convex polygons when one of them moves to another along a given direction.First,several novel concepts and proprties are explored,then an optimal algorithm OPFIV with time complexity O(log n m)) is developed and its correctness and optimization are proved rigorously.     

A convex hull algorithm for planar simple polygons

A linear convex hull algorithm which is an improvement on the algorithm due to Sklansky is given.     

A simple linear algorithm for intersecting convex polygons

LetP andQ be two convex polygons withm andn vertices, respectively, which are specified by their cartesian coordinates in order. A simpleO(m+n) algorithm is presented for computing the intersection ofP andQ. Unlike previous algorithms, the new algorithm consists of a two-step combination of two simple algorithms for finding convex hulls and triangulations of polygons.     

A simple linear algorithm for intersecting convex polygons

LetP andQ be two convex polygons withm andn vertices, respectively, which are specified by their cartesian coordinates in order. A simpleO(m+n) algorithm is presented for computing the intersection ofP andQ. Unlike previous algorithms, the new algorithm consists of a two-step combination of two simple algorithms for finding convex hulls and triangulations of polygons.     

A simple linear hidden-line algorithm for star-shaped polygons

A very simple, linear-running-time algorithm is presented for solving the hidden-line problem for star-shaped polygons. The algorithm first decomposes the visibility regions into edge-visible polygons and then solves the hidden-line problem for these simpler polygons. In addition to simplicity the algorithm possesses the virtue of affording a very easy proof of correctness. Some applications where this problem arises are mentioned.     

An algorithm for recognizing palm polygons

A polygonP is said to be apalm polygon if there exists a pointxP such that the Euclidean shortest path fromx to any pointyP makes only left turns or only right turns. The set of all such pointsx is called thepalm kernel. In this paper we propose an O(E) time algorithm for recognizing a palm polygonP, whereE is the size of the visibility graph ofP. The algorithm recognizes the given polygonP as a palm polygon by computing the palm kernel ofP. If the palm kernel is not empty,P is a palm polygon.The extended abstract of this paper was reported at the Second Canadian Conference in Computational Geometry, pp. 246–251, 1990     

An optimal algorithm for computing the minimum vertex distance between two crossing convex polygons

LetP={p ₁ ,p ₂ , ...,p _m } andQ={q ₁ ,q ₂ , ...,q _n } be two intersecting convex polygons whose vertices are specified by their cartesian coordinates in order. An optimalO(m+n) algorithm is presented for computing the minimum euclidean distance betweena vertexp _i inP and a vertexq _j inQ.     

Numerical stability of a convex hull algorithm for simple polygons

A numerically stable and optimalO(n)-time implementation of an algorithm for finding the convex hull of a simple polygon is presented. Stability is understood in the sense of a backward error analysis. A concept of the condition number of simple polygons and its impact on the performance of the algorithm is discussed. It is shown that if the condition number does not exceed (1+O())/(3), then, in floating-point arithmetic with the unit roundoff, the algorithm produces the vertices of a convex hull for slightly perturbed input points. The relative perturbation does not exceed 3(1+O()).J. W. Jaromczyk was partially supported by a grant from the Center for Robotics and Manufacturing Systems at the University of Kentucky and G. W. Wasilkowski was partially supported by the National Science Foundation under Grants CCR-89-05371 and CCR-91-14042.     

A simple algorithm for the unique characterization of convex polygons

A simple method for specifying the shape and orientation of a convex polygon is described. The method utilizes eigenvalues and eigenvectors of a “moment of inertia” or mass matrix computed from the nodal coordinates of the polygon. The “shape” is characterized then by the parameter ( , where ξ₁ and ξ₂ are the eigenvalues (ξ₁ ξ₂), and the orientation by the axial direction of the first eigenvector. FORTRAN subroutines are provided for this algorithm.     

A simple nc algorithm to recognize weakly triangulated graphs

Let S be a set of n closed intervals on the x-axis. A ranking assigns to each interval, s, a distinct rank, p(s)? [1, 2,…,n]. We say that s can see t if p(s)<p(t) and there is a point p?s∩t so that p?u for all u with p(s)<p(u)<p(t). It is shown that a ranking can be found in time O(n log n) such that each interval sees at most three other intervals. It is also shown that a ranking that minimizes the average number of endpoints visible from an interval can be computed in time O(n ^5/2). The results have applications to intersection problems for intervals, as well as to channel routing problems which arise in layouts of VLSI circuits.     

An algorithm for triangulating multiple 3D polygons

We present an algorithm for obtaining a triangulation of multiple, non‐planar 3D polygons. The output minimizes additive weights, such as the total triangle areas or the total dihedral angles between adjacent triangles. Our algorithm generalizes a classical method for optimally triangulating a single polygon. The key novelty is a mechanism for avoiding non‐manifold outputs for two and more input polygons without compromising optimality. For better performance on real‐world data, we also propose an approximate solution by feeding the algorithm with a reduced set of triangles. In particular, we demonstrate experimentally that the triangles in the Delaunay tetrahedralization of the polygon vertices offer a reasonable trade off between performance and optimality.     

A parallel algorithm for constructing reduced visibility graph and its FPGA implementation

A central geometric structure in applications such as robotic path planning and hidden line elimination in computer graphics is the visibility graph. A new parallel algorithm to construct the reduced visibility graph in a convex polygonal environment is presented in this paper. The computational complexity is O(p²log(n/p)) where p is the number of objects and n is the total number of vertices. A key feature of the algorithm is that it supports easy mapping to hardware. The algorithm has been simulated (and verified) using C. Results of hardware implementation show that the design operates at high speed requiring only small space. In particular, the hardware implementation operates at approximately 53 MHz and accommodates the reduced visibility graph of an environment with 80 vertices in one XCV3200E device.