Worst-case optimal hidden-surface removal
ACM Transactions on Graphics (TOG)
Art gallery theorems and algorithms
Art gallery theorems and algorithms
Boundary evaluation and direct display of CSG models
Computer-Aided Design
Hybrid shadow testing scheme for ray tracing
Computer-Aided Design
An algorithmic approach to some problems in terrain navigation
Geometric reasoning
Triangulation and shape-complexity
ACM Transactions on Graphics (TOG)
An algorithm for planning collision-free paths among polyhedral obstacles
Communications of the ACM
Effective Computational Geometry for Curves and Surfaces (Mathematics and Visualization)
Effective Computational Geometry for Curves and Surfaces (Mathematics and Visualization)
Guard placement for efficient point-in-polygon proofs
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Guarding curvilinear art galleries with vertex or point guards
Computational Geometry: Theory and Applications
Guarding curvilinear art galleries with vertex or point guards
Computational Geometry: Theory and Applications
Computational Geometry: Theory and Applications
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In this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves. We consider two types of guards: edge guards (these are edges of the polygon) and mobile guards (these are either edges or straight-line diagonals of the polygon). Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs. We reduce the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a constrained triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in the 2-dominating set. We show that, given a triangulation graph Tp of a polygon P with n ≥ 3 vertices: (1) ⌊n+1/3⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊2n+1/5⌋ edge guards are always sufficient and ⌊2n/5⌋ edge guards are sometimes necessary, in order to 2-dominate Tp. We also show that a diagonal (resp., edge) 2-dominating set of size ⌊2n+1/5⌋ can be computed in O(n2) time and O(n) space. Based on these results we prove that, in order to monitor a piecewise-convex polygon P witn n ≥ 2 vertices: (1) ⌊n+1/3⌋ mobile guards or ⌊2n+1/5⌋ edge guards are always suffcient, and (2) ⌊n/3⌋ mobile edge guards are sometimes necessary. A mobile (resp., edge) guard set for P of size ⌊n+1/3⌋ (resp., ⌊2n+1/5⌋ or ⌊3n/7⌋ can be computed in O(n log n + T(n)) time and O(n) space, where T(n) denotes the time for computing a diagonal (resp., edge) 2-dominating set of size ⌊n+1/3⌋ (resp., ⌊2n+1/5⌋ or ⌊3n/7⌋ for a trinagulation graph with n vertices.