H. ElGindy and D. Avis. A linear algorithm for computing the visibility of polygon from a point. J. Algorithms, 2:186--197, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... has been studied by Chv atal [12] Aggarwal [1] O Rourke [26] and Shermer [30] The visibility polygon problem (where, given a polygon P and a point q inside the polygon, determine the polygon consisting of all the points in P which are visible from q) has been studied by ElGindy and Avis [16], Lee [22] Joe and Simpson [20] O Rourke s book [26] surveys many visibility problems which have been investigated. In this thesis, we consider several questions concerning visibility, all stemming from the following question: Given a query point q in the interior of a simple polygon, P , find ....

....level description of the preprocessing step as described in the overview of Chapter 1. 1. Construct the set W (as defined in chapter 3) ffl The set of windows with respect to one vertex in polygon P can be constructed in O(n) time using a standard visibility polygon algorithm (ElGindy and Avis [16], Lee [22] Joe and Simpson [20] This implies ) complexity for this step. 2. Construct the planar subdivision induced by this set. ffl Although there is an algorithm by Chazelle and Edelsbrunner[11] which finds the planar subdivision induced by a set of line segments in O(g log g i) time ....

H. ElGindy and D. Avis. A Linear Algorithm for Computing the Visibility Polygon from a Point. Journal of Algorithms, 2, pp. 186-197, 1981.

....O Rourke [27] and Shermer [31] A survey paper by Shermer [32] reviews the recent results in this area. The visibility polygon problem (given a polygon P and a point q inside the polygon, determine the polygon consisting of all the points in P visible from q) has been studied by ElGindy and Avis [16], Lee [23] Joe and Simpson [21] O Rourke s book [27] surveys many visibility problems which have been investigated. We consider several questions concerning visibility all stemming from the following question: Given a query point q in the interior of a simple polygon P nd all the vertices of P ....

....we give a high level description of the preprocessing step as described in Section 1. 1. Construct the set W (as de ned in Section 3) The set of windows with respect to one vertex in polygon P can be constructed in O(n) time using a standard visibility polygon algorithm (ElGindy and Avis [16], Lee [23] Joe and Simpson [21] This implies O(n ) complexity for this step. 2. Construct the planar subdivision induced by this set. Although there is an algorithm by Chazelle and Edelsbrunner[14] which nds the planar subdivision induced by a set of line segments in O(g log g i) time ....

H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. Journal of Algorithms, 2, pp. 186-197, 1981.

....intersecting or non intersecting line segments, simple polygons, planes, etc. Applications of the visibility problems include the art gallery and illumination [1] minimum distance between Steiner trees [5] the skyline problem [6] etc. Two algorithms performing in linear time are described in [4] for a domain of segments in the plane that form a single simple polygon. In [9] various special types of visibility problems are explored through the definition of set theoretic operations on visibility polygons. An algorithm that finds the visibility polygon of a point inside a polygon is used ....

H. ElGindy and D. Aris, " A Linear Algorithm for Computing the Visibility Polygon from a Point", J. Algorithms 2(2), p.p. 186-197, 1981

....for intersections. We start by computing the polygon consisting of all points visible from the endpoint e of the last edge in the convex direction. We call this polygon the visibility polygon, as illustrated in Figure 16. This polygon is computable in O(n) time by an algorithm of Avis and El Gindy [11], but is more easily computed with the help of the following observation. Because the chain consists of all right turns, the visibility polygon consists of a chain of re ex vertices (of edges 1 and 2 in Figure 16) in the middle of the chain, followed by a chain of convex vertices (of edges 3 ....

H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms, 2:186-197, 1981. 16

....2. Description of the algorithm. In this section we will describe the algorithm for computing the visibility polygon of a point p inside a simple polygon P . The first linear time algorithm for computing the visibility polygon of a point inside a simple polygon was described by ElGindy and Avis [1]. Lee [4] and later Joe and Simpson [2] have simplified the algorithm. Our algorithm is loosely connected to the one of Joe and Simpson. In chapter 8 of the book by O Rourke [6] also more general visibility algorithms are presented, for example the visibility polygon of an edge e inside a polygon. ....

H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms, 2:186--197, 1981.

....A chain which is spiralling outward. Figure 14: A chain which is spiralling inward. from the endpoint e of the last edge in the convex direction. We call this polygon the visibility polygon, as illustrated in Figure 15. This polygon is computable in O(n) time by an algorithm of Avis and El Gindy [11], but is more easily computed with the help of the following observation. Because the chain consists of all right turns, the visibility polygon consists of a chain of reflex vertices (of edges 1 and 2 in Figure 15) in the middle of the chain, followed by a chain of convex vertices (of edges 3 ....

H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms, 2:186--197, 1981.

....simple polygons have been extensively studied in the last two decades. Actually, they are in the core of the origins of Computational Geometry. There are optimal algorithms for computing the shortest path tree from a point (Guibas et al. 12] the visibility polygon from a point (ElGindy and Avis [9] and Lee [14] the weak visibility polygon from an edge (Guibas et al. 12] Lee and Lin [15] and Toussaint [21] There are also optimal algorithms to decide if a polygon is star shaped (Lee and Preparata [16] to decide if a polygon is weakly visible from an edge (Avis and Toussaint [3] and ....

H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms, 2:186--197, 1981.

....them. Since each vertex is projected at most three times, we thus increase the number of unprotected edges by less than 12n. Here is how we find the at most three unprotected edges for a reflex vertex c. The part of visible from c can be computed in a single walk along the boundary of , see e.g. [ElAv81, Lee83, JoSi87]. The amount of time needed for the walk is proportional to the number of edges. Select the at most three edges that have connected portions visible from c along an angle exceeding 2 . Project c orthogonally onto these at most three edges. Each projection subdivides an unprotected edge into two ....

H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms 2 (1981), 186--197.

....for p can be done in time linear in the number of vertices. It follows that quadratic time suffices for step 1. Computing the sorted sequence of diagonals pp 1 ; pp 2 ; pp m incident to p is a standard operation for simple polygons and can be done in linear time; see, for example, [ElAv81, JoSi87, Lee83]. Let pp 0 and pp m 1 be the two edges of R incident to p and assume that p 0 ; p 1 ; p 2 ; p m 1 is in a counterclockwise order around p. To determine whether there is a vertex of R in the half lune j pp i for 1 i m, we scan the list p 0 ; p 1 ; p m 1 once, from smallest ....

....projected at most three times, we thus increase the number of unprotected edges by less than 12n. Here is how we find the at most three unprotected edges for a reflex vertex c. The parts of the boundary of visible from c can be computed in a single walk along the boundary of ; see, for example, [ElAv81, Lee83, JoSi87]. The amount of time needed for the walk is proportional to the number of edges. Select the at most three edges that have connected portions visible from c along an angle exceeding 2 . Project c orthogonally onto these at most three edges. Each projection subdivides an unprotected edge into two ....

H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms 2 (1981), 186--197.

....types of guards, with different powers of vision. The visibility polygon with respect to a given point inside a simple polygon is the set of points that are directly visible to that point. Linear time algorithms for constructing point visibility polygons were presented by ElGindy and Avis [8], by Lee [16] and by Guibas et al. 10] Horn and Valentine introduced the concept of k link visibility, defining a point D to be k link visible to a point S if there exists a k link polygonal path between them that remains inside the 1 Department of Computer and Information Science, ....

....This give an Omega Gamma n 2 ) lower bound on the worst case complexity of Vd(S) We now derive an upper bound. We begin by recalling that Vd(S) is the union of V (S) and the at most n mirror visibility polygons. Each of the at most n 1 sets is a polygonal set cut out of P by at most n chords [8, 16, 10]. Thus only O(n 2 ) vertices can occur on the boundary of P . Only O(n 2 ) vertices can occur on windows of V (S) at most two per edge of the relative boundary of a mirror visibility polygon. Vd(S) is a polygon whose interior vertices can only be left, right, or top vertices 19 of boundary ....

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H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms, 2:186--197, 1981.

....been the prime source of many problems in this area. Two points inside a simple polygon are mutually visible if the line segment joining them is not obstructed by any edge of the polygon. Several algorithms for computing the region visible from a point light source inside a simple polygon exist [5, 9, 7]. The computation of the region of a simple polygon which is weakly visible from an internal segment [10, 7] or a convex set [6] are also well studied. Certain portions of the polygon that are not directly illuminated from the source may become visible due to one or more reflections on the ....

....y is 2 visible from S; y is visible from S after one diffuse reflection. We assume that the light incident at a vertex is absorbed and not reflected further. Let V (S) denote the portion of P visible from S. We know that V (S) has O(n) edges and at most one edge of V (S) lies on each edge of P [5, 7]. For k 0, let V d k (S) denote the polygonal region consisting of points that are l visible from S, for some 1 l k 1. Informally, V d k (S) is the set of points that receive light from S after at most k diffuse reflections. We have V d 0 (S) V (S) We define a mirror at the kth stage of ....

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H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. Journal of Algorithms, 2:186--197, 1981.

....paper, we feel that the problem might be hard even in this simple context. We do not address this question directly, however. Direct visibility has been investigated extensively over the past several years, and a number of linear time direct visibility algorithms for simple polygons are known [10, 14]. Among different alternative notions of visibility, k link visibility comes closest to what we study in this paper. Horn and Valentine introduced this concept, where a point y inside a given polygon is k link visible from another point x if there exists a k link polygonal path between them inside ....

....S with respect to the mirror m i . Then considering the direct visibility polygon of S i in the polygon P [ T , T being the triangle formed by S i and m i , we can obtain V i as described in lemma 3.1. Direct visibility polygons can be computed by any one of the known linear time algorithms [10, 14]. More precisely, it is easy to check that a triangulation based algorithm will work correctly on the Riemann surface P [ T , even if, as subsets of the plane, P and T overlap. After computing all V i s we apply a divide and conquer technique to compute the final visibility polygon V k (S) Let ....

H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms, 2:186--197, 1981.

....to some forward essential cut in FEC i . This concludes the proof. 2 The loop of Step 3 is performed k times. Each time the limit point of the previous vision point is computed in O(n log(n m) time as follows: Let g i be the previously computed vision point. Compute VP(g i ) in O(n) time [10, 18, 21]. For each pocket in VP(W (g s ; g i ) identify the two end points q and q 0 of the associated window. Let q be the convex vertex of VP(W (g s ; g i ) on the window and let q 0 be the other end point; see Figure 11. We handle each of the three cases of Lemma 4.8 separately. 1. Given the ....

H. ElGindy, D. Avis. A Linear Algorithm for Computing the Visibility Polygon from a Point. J. Algorithms, 2:186197, 1981.

....diagonal p i p j with i j. By convention, we will say that d 0(n Gamma1) is also a diagonal, and sometimes we will allow d ij to denote a polygon edge when j = i 1. Diagonals for a given vertex p i can be found by computing the visibility polygon for p i , which can be done in linear time [6]. The following observation is important for visibility algorithms, and will be important for us. Observation 1 The order of diagonals ccw around a vertex p i is the same as the ordering of their other endpoints ccw around P . A vertex of P is reflex if its interior angle is greater than . It ....

H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms, 2:186--197, 1981.

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H. ElGindy and D. Avis. A linear algorithm for computing the visibility of polygon from a point. J. Algorithms, 2:186--197, 1981.

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H. A. E. GINDY AND D. AVIS, A linear algorithm for computing the visibility polygon from a point, J. Algorithms, 2 (1981), pp. 186--197.

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H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms, 2:186197, 1981.

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H. ElGindy and D. Avis. A linear algorithm for computing the visibility polygon from a point. J. Algorithms, 2:186--197, 1981.

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H. ElGindy and D. Avis, "A linear algorithm for computing the visibility polygon from a point" J. Algorithms 2, 186-197 (1981).

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