Computational complexity of art gallery problems
IEEE Transactions on Information Theory
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
Triangulating a simple polygon in linear time
Discrete & Computational Geometry
Decomposing polygonal regions into convex quadrilaterals
SCG '85 Proceedings of the first annual symposium on Computational geometry
Triangulation and shape-complexity
ACM Transactions on Graphics (TOG)
An algorithm for planning collision-free paths among polyhedral obstacles
Communications of the ACM
Rectilinear computational geometry
Rectilinear computational geometry
The art gallery theorem: its variations, applications and algorithmic aspects
The art gallery theorem: its variations, applications and algorithmic aspects
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 edge or mobile guards
Proceedings of the 2008 ACM symposium on Solid and physical modeling
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
A combinational approach to polygon similarity
IEEE Transactions on Information Theory
Guarding curvilinear art galleries with edge or mobile guards
Proceedings of the 2008 ACM symposium on Solid and physical modeling
Computational Geometry: Theory and Applications
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We study a variant of the classical art gallery problem, where an art gallery is modeled by a polygon with curvilinear sides. We focus on piecewise-convex and piecewise-concave polygons, which are polygons whose sides are convex and concave arcs, respectively. It is shown that for monitoring a piecewise-convex polygon with n=2 vertices, @?2n3@? vertex guards are always sufficient and sometimes necessary. We also present an algorithm for computing at most @?2n3@? vertex guards in O(nlogn) time and O(n) space. For the number of point guards that can be stationed at any point in the polygon, our upper bound @?2n3@? carries over and we prove a lower bound of @?n2@?. For monitoring a piecewise-concave polygon with n=3 vertices, 2n-4 point guards are always sufficient and sometimes necessary, whereas there are piecewise-concave polygons where some points in the interior are hidden from all vertices, hence they cannot be monitored by vertex guards. We conclude with bounds for some special types of curvilinear polygons.