Some Problems Related to Good Illumination

Abellanas, Manuel; Bajuelos, Antonio L.; Matos, Inês

doi:10.1007/978-3-540-74472-6_1

Cited by 5 publications

(13 citation statements)

References 10 publications

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“…11, left). Box B x has necessarily to be large enough to guarantee, that each intersection point lies (1) outside the union of the wanted cones and (2) inside the intersection of the deepest cones of the given slope. The existence of box B x with these properties follows from the discussion above.…”

Section: Observationmentioning

confidence: 99%

“…Any set S of n points in the plain can be partitioned into O (r) disjoint classes by a simplicial partition, such that every simplex (i.e. triangle) contains between n r and 2n r points and every line crosses at most O (r 1 2 ) simplices (crossing number). Moreover, for any ξ > 0 such a simplicial partition can be constructed in O (n 1+ξ ) time.…”

Section: Algorithmmentioning

confidence: 99%

“…They further show that the unoriented π 2 -maxima can be computed in O(n) expected time. Abellanas et al [1] extent the guarding model (there it is called good Θillumination) by a range r, i.e., a guard g ∈ G can only guard points inside the circle of radius r that is centered at g. Beneath other results they show how to check if a query point p is Θ-guarded in O(n) time and output the necessary range and guards as witnesses.…”

Section: Previous Workmentioning

confidence: 99%

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Finding the Θ-guarded region

Matijević

Osbild

2010

Computational Geometry

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Convex hull generalizationGood Θ-illumination α-embracing contour We are given a finite set of n points (guards) G in the plane R 2 and an angle 0 Θ 2π . A Θ-cone is a cone with apex angle Θ. We call a Θ-cone empty (with respect to G) if it does not contain any point of G. A point p ∈ R 2 is called Θ-guarded if every Θ-cone with its apex located at p is non-empty. Furthermore, the set of all Θ-guarded points is called the Θ-guarded region, or the Θ-region for short.We present several results on this topic. The main contribution of our work is to describe the Θ-region with O ( n Θ ) circular arcs, and we give an algorithm to compute it. We prove a tight O (n) worst-case bound on the complexity of the Θ-region for Θ π 2 . In case Θ is bounded from below by a positive constant, we prove an almost linear bound O (n 1+ε ) for any ε > 0 on the complexity. Moreover, we show that there is a sequence of inputs such that the asymptotic bound on the complexity of their Θ-region is Ω(n 2 ).

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Section: Observationmentioning

confidence: 99%

Section: Algorithmmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

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Finding the Θ-guarded region

Matijević

Osbild

2010

Computational Geometry

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“…Let S be the set of n lights in the plane. We use an algorithm for computing the closest embracing site of a point introduced in [3] that runs in linear time and will be called "CES algorithm. "…”

Section: Algorithm For Computing the E-voronoi Diagrammentioning

confidence: 99%

“…Since there can be more than one embracing site per point, we want to compute the closest embracing site for q, which can be done in a linear time [3,7]. Note that there are points that have more than one closest embracing site though that is not the situation for most points.…”

Section: Introduction and Related Workmentioning

confidence: 99%

The embracing Voronoi diagram and closest embracing number

Abellanas¹,

Peñalver²,

Bajuelos

et al. 2009

J Math Sci

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A point q is embraced by a set of points S if q is interior to the convex hull of S [8]. In some illumination applications where points of S are lights and q is a point to be illuminated, the embracing concept is related to a good illumination [4,6], also known as the -guarding [12] and the well-covering [10]. In this paper, we are not only interested in convex dependency (which is actually the embracing notion) but also in proximity. Suppose that the sites of S are lights or antennas with limited range; due to their limited power, they cover a disk of a given radius r centered at the sites of S. Only the points lying in such disks are illuminated. If we want to embrace the point q with the minimum range r, we need to know which is the closest light sq to q such that q lies in the convex hull formed by sq and the lights of S closer to q than sq. This subset of S related to point q is called the closest embracing set for q in relation to S and its cardinality is the closest embracing number of q. By assigning every point q in the convex hull of S to its closest embracing site sq, we obtain a partition of the convex hull that we call the embracing Voronoi diagram of S. This paper proves some properties of the embracing Voronoi diagrams and describes algorithms to compute such diagrams, as well as the levels in which the convex hull is decomposed regarding the closest embracing number.

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References

2013

Stochastic Geometry and Its Applications

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Some Problems Related to Good Illumination

Cited by 5 publications

References 10 publications

Finding the Θ-guarded region

Finding the Θ-guarded region

The embracing Voronoi diagram and closest embracing number

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