Visibility Polygon Research Articles

THE VISIBILITY COMPLEX

We introduce the visibility complex (a 2-dimensional regular cell complex) of a collection of n pairwise disjoint convex obstacles in the plane. It can be considered as a subdivision of the set of free rays (i.e., rays whose origins lie in free space, the complement of the obstacles). Its cells correspond to collections of rays with the same backward and forward views. The combinatorial complexity of the visibility complex is proportional to the number k of free bitangents of the collection of obstacles. We give an O(n log n+k) time and O(k) working space algorithm for its construction. Furthermore we show how the visibility complex can be used to compute the visibility polygon from a point in O(m log n) time, where m is the size of the visibility polygon. Our method is based on the notions of pseudotriangle and pseudo-triangulation, introduced in this paper.

Formula omitted][formula omitted]-Algorithms for Minimum Link Path and Related Problems

The link metric, defined on a constrained region R of the plane, sets the distance between a pair of points in R to equal the minimum number of line segments or links that are needed to construct a path in R between the points. The minimum link path problem is to compute a path consisting of the minimum number of links between two points in R, when R is the inside of an n-sided simple polygon. The minimum nested polygon problem asks for a minimum link closed path (girth) when R is an annular region defined by a pair of nested simple polygons. Efficient sequential algorithms based on greedy methods have been described for both problems. However, neither problem was known to be N C . In this paper we present algorithms that require O(log n loglog n) time and O( n) space using O( n) processors for both problems. The approach used involves new results on the parallel ( N C 1) computation of the complete visibility polygon of a simple polygon from a set of points inside it, along with an algebraic technique based on fractional linear transforms that permits effective parallelization of the "greedy" computations. The complexity results of this paper are with respect to the CREW-PRAM model of computation. The time × processor product of these algorithms is within a small polylog factor of the best known sequential algorithms for the respective problems.

An optimal algorithm for finding the edge visibility polygon under limited visibility

Abstract Two points in a polygon P are said to be d-visible if the line segment l joining them is contained in P and the length of l is d or less. In applications such as computer graphics and robot vision, the notion of d -visibility is more suitable than the commonly used definition of visibility, which is equivalent to d -visibility with infinite d . We present an O( n ) time algorithm for finding the d -visibility polygon from an edge e in P , where n is the number of vertices in P .

An Optimal Algorithm for Computing Visibility in the Plane

The authors give an algorithm to compute the visibility polygon from a point among a set of $h$ pairwise-disjoint polygonal obstacles with a total of $n$ vertices. Our algorithm uses $O(n)$ space and runs in optimal time $\Theta(n+h\log h)$, improving the previous upper bound of $O(n\log n)$. A direct consequence of the algorithm is an $O(n+h\log h)$ time algorithm for computing the convex hull of $h$ disjoint simple polygons.

The Superman problem

Given a polygonK contained in a polygonP, and a points lying outsideP, we present a Θ (n logn) algorithm that finds the minimum number of edges, ofP that we want to retain in order to hidek froms. Furthermore, if the visibility polygon ofs givenK is unbounded, the algorithm is shown to run in linear time. This paper is dedicated to J. Siegel and J. Shuster.

On the number of guard edges of a polygon

This paper establishes tight bounds on the number of edges of a polygon from which every point in the polygon is visible; we call them guard edges. For a nonstarshaped polygon, there can be at most three guard edges. For a polygon with holes, there may be at most six; three on the outer boundary and three on one of the holes. The results give new insights into the structure of visibility in polygons and shed light on developing an efficient algorithm for finding all guard edges of a polygon with or without holes.

Characterizing and recognizing weak visibility polygons

A polygon is said to be a weak visibility polygon if every point of the polygon is visible from some point of an internal segment. In this paper we derive properties of shortest paths in weak visibility polygons and present a characterization of weak visibility polygons in terms of shortest paths between vertices. These properties lead to the following efficient algorithms: (i) an O( E) time algorithm for determining whether a simple polygon P is a weak visibility polygon and for computing a visibility chord if it exist, where E is the size of the visibility graph of P and (ii) an O( n 2) time algorithm for computing the maximum hidden vertex set in an n-sided polygon weakly visible from a convex edge.

Determination of minimum number of sensors and their locations for an automated facility: An algorithmic approach

We present an algorithmic approach to determine the minimum number of sensors and their placements required to monitor the workplace. For a 2-D workspace this involves finding the minimal sensor placement to ‘see’ the edges of all the objects (polygons) in the workspace and the boundary of the workspace. The problem belongs to a class of geometric optimization problems called the art gallery problems and is a variant of the minimum guard cover problem (MGCP). Utilizing the concepts of visibility of a point, we characterize the problem of strong visibility of an edge in a polygon with holes. An efficient algorithm is presented which runs in time linear to the number of edges present to obtain the strong visibility polygon for an edge. The visibility polygons of all edges give us a set of points which is sufficient to include the optimal placement of sensors. A set covering problem is then solved on this set of points to extract a minimal set of points that provides visibility of all the desired edges. The set covering problem is NP-complete and makes the overall approach NP-complete (the MGCP is also known to be NP-complete). However, the finite set of sufficient location points and efficient heuristics for the set covering problem make this method a viable one.

A scan conversion algorithm using quad‐tree representation of dZ buffer

AbstractThis paper proposes an extension of the Z buffer or dZ buffer that stores not only the Z component but also the normal vector of a visible polygon at the sampling point. While the conventional Z buffer algorithm is based on a zeroth‐order approximation of a visible polygon, the dZ buffer is considered as a first‐order approximation. As a result, the dZ buffer can approximate a surface by fewer sampling points.A quad‐tree based internal representation of the dZ buffer and scan conversion algorithm has been proposed using the representation. The proposed algorithm uses memory in proportion to the complexity of the generated image while the Z buffer uses memory in proportion to the image size. While the algorithm allows an incremental update of the dZ buffer as similarly to the Z buffer, anti‐alias processing is possible by controlling oversampling on demand.The algorithm can be implemented using recursive procedure calls including simple overlapping detection of triangles and rectangles. The algorithm is implemented and evaluated on a UNIX workstation and the result shows better figures in both conversion time and memory requirement.

GEOMPACK — a software package for the generation of meshes using geometric algorithms

We describe the mathematical software package GEOMPACK, which contains standard Fortran 77 routines for the generation of two-dimensional triangular and three-dimensional tetrahedral finite element meshes using efficient geometric algorithms. This package results from our research into mesh generation and geometric algorithms. It contains routines for constructing two- and three-dimensional Delaunay triangulations, decomposing a general polygonal region into simple or convex polygons, constructing the visibility polygon of a simple polygon from a viewpoint, and other geometric algorithms, from which our mesh generation method is built and others can be implemented. Our method generates meshes in polygonal or polyhedral regions specified by their boundary representation and possible interfaces between subregions.

Computing the visibility polygon from a convex set and related problems

In this paper, we propose efficient algorithms for computing the complete and weak visibility polygons of a simple polygon P of n vertices from a convex set C inside P. The algorithm for computing the complete visibility polygon of P from C takes O( n + k) time in the worst case, where k is the number of extreme points of the convex set C. Given a triangulation of P - C, the algorithm for computing the weak visibility polygon of P from C takes O( n + k) time in the worst case. We also show that computing the complete and weak visibility polygons of P from a nonconvex set inside P has the same time complexity. The algorithm for computing the complete visibility polygon of P from a convex set inside P leads to efficient algorithms for the following problems: (i) Given a polygon Q of m vertices inside another polygon P of n vertices, construct a minimum nested convex polygon K between P and Q. The algorithm runs in O(( n + m)log k) time, where k is the number of vertices of K. This is an improvement over the O(( n + m)log( n + m)) time algorithm of Wang and Chan. (ii) Given two points inside a polygon P, compute a minimum link path between them inside P. Given a triangulation of P, the algorithm takes O( n) time. Suri also proposed a linear time algorithm for this problem in a triangulated polygon but our algorithm is simpler.

Fitting polygonal functions to a set of points in the plane

Given a set of points S = {( x 1, y 1), ( x 2, y 2), …, ( x N , y N )} in R 2 with x 1 < x 2 < … < x N , we want to construct a polygonal (i.e., continuous, piecewise linear) function f with a small number of corners (i.e., nondifferentiable points) which fits S well. To measure the quality of f in this regard, we employ two criteria: 1. (i) the number of corners in the graph of f, and 2. (ii) max 1≤i≤N¦y i − f(x i)¦ (the Chebyshev error of the fit). We give efficient algorithms to construct a polygonal function f that minimizes (i) (resp. (ii)) under a maximum allowable value of (ii) (resp. (i)), whether or not the comers of f are constrained to be in the set S. A key tool used in designing these algorithms is a linear time algorithm to find the visibility polygon from an edge in a monotone polygon. A variation of one of these algorithms solves the following computational geometry problem in optimal O( N) time: Given N vertical segments in the plane, no two with the same abscissa, find a monotone polygonal curve with the least number of corners which intersects all the segments.

On the correctness of a linear-time visibility polygon algorithm∗

We prove the correctness of the linear-time algorithm in Joe and Simpson [7] for computing the visibility polygon of a simple polygon from a viewpoint that is either interior to the polygon or in its blocked exterior. This proof establishes pre- and post-conditions for all steps of the algorithm. The algorithm and the proof are identical for interior and blocked exterior viewpoints. Other algorithms for computing the visibility polygon (usually for interior viewpoints only) are either nonlinear-time and harder to implement or can fail for polygons that wind sufficiently.

On geodesic properties of polygons relevant to linear time triangulation

Triangulating a simple polygon ofn vertices inO(n) time is one of the main open problems in computational geometry. The fastest algorithm to date, due to Tarjan and van Wyk, runs inO(n log logn), but several classes of simple polygons have been shown to admit linear time traingulation. Famous examples of such classes are: star-shaped, monotone, spiral, edge visible, and weakly externally visible polygons. The notion of geodesic paths is used here to characterize all classes of polygons for which linear time triangulation algorithms are known. First we introduce a new class of polygons,palm polygons, which subsumes many known classes of polygons for which linear time triangulation algorithms exist, and present a linear time algorithm for triangulating polygons in this class. Then a class of polygons,crab polygons, is defined and shown to contain all classes of existing polygons for which linear time triangulation algorithms are known. As a byproduct of this characterization, a new, very simple linear time algorithm for triangulating star-shaped polygons is obtained.

Systolic algorithms for computing the visibility polygon and triangulation of a polygonal region

This paper first presents a naive systolic algorithm for finding a closest point for each of n given points in linear time. Then, based on the algorithm, we propose linear-time systolic algorithms for the computation of the visibility polygon and for the trapezoidal partition or triangulation of a polygonal region which may contain holes. The visibility problem among n vertical line segments in the plane is also solved.

Corrections to Lee's visibility polygon algorithm

We present a modification and extension of the (linear time) visibility polygon algorithm of Lee. The algorithm computes the visibility polygon of a simple polygon from a viewpoint that is either interior to the polygon, or in its blocked exterior (the cases of viewpoints on the boundary or in the free exterior being simple extensions of the interior case). We show by example that the original algorithm by Lee, and a more complex algorithm by El Gindy and Avis, can fail for polygons that wind sufficiently. We present a second version of the algorithm, which does not extend to the blocked exterior case.

Shortest path between two simple polygons

Consider a collection of mutually disjoint simple polygons in the plane containing a total of n edges. Two of them are specified as a source polygon S and a target polygon T. We present an efficient algorithm for finding a shortest path between S and T avoiding the other polygons. We show that it runs in O(n 2) time, using a linear-time algorithm for computing the visibility polygon of a point. This problem is related to a wire routing design of a certain type of LSI for which terminals are of polygonal shape and larger than a wire segment.

Visibility of disjoint polygons

Consider a collection of disjoint polygons in the plane containing a total ofn edges. We show how to build, inO(n 2) time and space, a data structure from which inO(n) time we can compute the visibility polygon of a given point with respect to the polygon collection. As an application of this structure, the visibility graph of the given polygons can be constructed inO(n 2) time and space. This implies that the shortest path that connects two points in the plane and avoids the polygons in our collection can be computed inO(n 2) time, improving earlierO(n 2 logn) results.

Computing the visibility polygon from an edge

We consider the problem of computing the visible region of a simple polygon from a line segment in the polygon. The region, called the visibility polygon, is the area that can be illuminated by a tubular light source represented as a line segment. We present an O(nlog n) algorithm for computing the visibility polygon, where n is the number of vertices of the polygon. The algorithm has been implemented in Pascal and some computer output samples are also included.

VLSI systems for some problems of computational geometry

Fast solution of the computational geometry problems is important for computer graphics, image processing and pattern recognition. The capability of the network Mesh of Trees for application in VLSI systems solving fastly the computational geometry problems is shown on two examples: determination of the convex hull of a weakly externally visible polygon and determination of the visibility polygon of a polygon.