Finding the Visibility Matrix of an Orthogonal Polygon

2012 ◽  
Author(s):  
Amrita Agarwala
Keyword(s):  
Orthogonal Polygon

scholarly journals An Upper Bound on the k-Modem Illumination Problem

2015 ◽  
Vol 25 (04) ◽  
pp. 299-308
Author(s):  
Frank Duque ◽  
Carlos Hidalgo-Toscano
Keyword(s):  
Upper Bound ◽  
Line Segment ◽  
Time Algorithm ◽  
Fixed Number ◽  
Light Sources ◽  
Wireless Devices ◽  
Orthogonal Polygon ◽  
Illumination Problem ◽  
Number Formula ◽  
Interior Points

A variation on the classical polygon illumination problem was introduced in [Aichholzer et al. EuroCG’09]. In this variant light sources are replaced by wireless devices called [Formula: see text]-modems, which can penetrate a fixed number [Formula: see text], of “walls”. A point in the interior of a polygon is “illuminated” by a [Formula: see text]-modem if the line segment joining them intersects at most [Formula: see text] edges of the polygon. It is easy to construct polygons of [Formula: see text] vertices where the number of [Formula: see text]-modems required to illuminate all interior points is [Formula: see text]. However, no non-trivial upper bound is known. In this paper we prove that the number of kmodems required to illuminate any polygon of [Formula: see text] vertices is [Formula: see text]. For the cases of illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we give a tighter bound of [Formula: see text]. Moreover, we present an [Formula: see text] time algorithm to achieve this bound.

Quadrilaterizing an Orthogonal Polygon in Parallel

1998 ◽  
Vol 44 (1) ◽  
pp. 50-68
Author(s):  
Jana Dietel ◽  
Hans-Dietrich Hecker
Keyword(s):  
Orthogonal Polygon

POLYGON DECOMPOSITION AND THE ORTHOGONAL ART GALLERY PROBLEM

2007 ◽  
Vol 17 (02) ◽  
pp. 105-138 ◽  
Author(s):  
CHRIS WORMAN ◽  
J. MARK KEIL
Keyword(s):  
Polynomial Time ◽  
Art Gallery ◽  
Minimum Cardinality ◽  
Decomposition Problem ◽  
Art Gallery Problem ◽  
Orthogonal Polygon ◽  
Polygon Decomposition

A decomposition of a polygon P is a set of polygons whose geometric union is exactly P. We study a polygon decomposition problem that is equivalent to the Orthogonal Art Gallery problem. Two points are r-visible if the orthogonal bounding rectangle for p and q lies within P. A polygon P is an r-star if there exists a point k ∈ P such that for each point q ∈ P, q is r-visible from k. In this problem we seek a minimum cardinality decomposition of a polygon into r-stars. We show how to compute the minimum r-star cover of an orthogonal polygon in polynomial time.

scholarly journals Orthogonal polygon reconstruction from stabbing information

2002 ◽  
Vol 23 (1) ◽  
pp. 69-83 ◽  
Author(s):  
L. Jackson ◽  
S.K. Wismath
Keyword(s):  
Orthogonal Polygon

Visibility concepts in orthogonal polygon recognition

2001 ◽  
Vol 22 (10) ◽  
pp. 1153-1159
Author(s):  
B Poorna ◽  
K.S Easwarakumar
Keyword(s):  
Orthogonal Polygon ◽  
Polygon Recognition

scholarly journals On the tileability of polygons with colored dominoes

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Chris Worman ◽  
Boting Yang
Keyword(s):  
Analysis Of Algorithms ◽  
Time Algorithm ◽  
Orthogonal Polygon ◽  
Orthogonal Polygons ◽  
Domino Tiling ◽  
International Audience ◽  
Np Complete

Analysis of Algorithms International audience We consider questions concerning the tileability of orthogonal polygons with colored dominoes. A colored domino is a rotatable 2 × 1 rectangle that is partitioned into two unit squares, which are called faces, each of which is assigned a color. In a colored domino tiling of an orthogonal polygon P, a set of dominoes completely covers P such that no dominoes overlap and so that adjacent faces have the same color. We demonstrated that for simple layout polygons that can be tiled with colored dominoes, two colors are always sufficient. We also show that for tileable non-simple layout polygons, four colors are always sufficient and sometimes necessary. We describe an O(n) time algorithm for computing a colored domino tiling of a simple orthogonal polygon, if such a tiling exists, where n is the number of dominoes used in the tiling. We also show that deciding whether or not a non-simple orthogonal polygon can be tiled with colored dominoes is NP-complete.

scholarly journals Pareto-Optimal Domino-Tiling of Orthogonal Polygon Phased Arrays

2021 ◽  
pp. 1-1
Author(s):  
Paolo Rocca ◽  
Nicola Anselmi ◽  
Alessandro Polo ◽  
Andrea Massa
Keyword(s):  
Phased Arrays ◽  
Pareto Optimal ◽  
Orthogonal Polygon ◽  
Domino Tiling

scholarly journals Orthogonal Art Galleries with Holes: A Coloring Proof of Aggarwal's Theorem

10.37236/1046 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Paweł Żyliński
Keyword(s):  
Arbitrary Number ◽  
Dual Graph ◽  
Art Galleries ◽  
Alternate Proof ◽  
Orthogonal Polygon ◽  
Orthogonal Polygons

We prove that $\lfloor{n+h\over 4}\rfloor$ vertex guards are always sufficient to see the entire interior of an $n$-vertex orthogonal polygon $P$ with an arbitrary number $h$ of holes provided that there exists a quadrilateralization whose dual graph is a cactus. Our proof is based upon $4$-coloring of a quadrilateralization graph, and it is similar to that of Kahn and others for orthogonal polygons without holes. Consequently, we provide an alternate proof of Aggarwal's theorem asserting that $\lfloor{n+h\over 4}\rfloor$ vertex guards always suffice to cover any $n$-vertex orthogonal polygon with $h \le 2$ holes.

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