Topological Art in Simple Galleries

Daniel Bertschinger; Nicolas  El Maalouly; Till Miltzow; Patrick Schnider; Simon Weber

doi:10.48550/arXiv.2108.04007

Topological Art in Simple Galleries

Daniel Bertschinger, Nicolas El Maalouly, Till Miltzow, Patrick Schnider, Simon Weber

Research output: Working paper › Preprint › Academic

Abstract

Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a,b∈P can see each other if the line segment seg(a,b) is contained in P. We denote by V(P) the family of all minimum guard placements. The Hausdorff distance makes V(P) a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set S there is a polygon P such that V(P) is homotopy equivalent to S.
Furthermore, for various concrete topological spaces T, we describe instances I of the art gallery problem such that V(I) is homeomorphic to T.

Original language	English
Publisher	arXiv
Pages	1-39
DOIs	https://doi.org/10.48550/arXiv.2108.04007
Publication status	Published - 9 Aug 2021

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title = "Topological Art in Simple Galleries",

abstract = "Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a,b∈P can see each other if the line segment seg(a,b) is contained in P. We denote by V(P) the family of all minimum guard placements. The Hausdorff distance makes V(P) a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set S there is a polygon P such that V(P) is homotopy equivalent to S.Furthermore, for various concrete topological spaces T, we describe instances I of the art gallery problem such that V(I) is homeomorphic to T.",

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AU - Schnider, Patrick

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AB - Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a,b∈P can see each other if the line segment seg(a,b) is contained in P. We denote by V(P) the family of all minimum guard placements. The Hausdorff distance makes V(P) a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set S there is a polygon P such that V(P) is homotopy equivalent to S.Furthermore, for various concrete topological spaces T, we describe instances I of the art gallery problem such that V(I) is homeomorphic to T.

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Topological Art in Simple Galleries

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