Topological Art in Simple Galleries
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Topological Art in Simple Galleries. / Bertschinger, Daniel; El Maalouly, Nicolas; Miltzow, Tillmann; Schnider, Patrick; Weber, Simon.
In: Discrete and Computational Geometry, Vol. 71, 2024, p. 1092–1130.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Topological Art in Simple Galleries
AU - Bertschinger, Daniel
AU - El Maalouly, Nicolas
AU - Miltzow, Tillmann
AU - Schnider, Patrick
AU - Weber, Simon
N1 - Publisher Copyright: © 2023, The Author(s).
PY - 2024
Y1 - 2024
N2 - 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.
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.
KW - Art gallery problem
KW - Computational geometry
KW - Topological universality
U2 - 10.1007/s00454-023-00540-x
DO - 10.1007/s00454-023-00540-x
M3 - Journal article
AN - SCOPUS:85169125034
VL - 71
SP - 1092
EP - 1130
JO - Discrete & Computational Geometry
JF - Discrete & Computational Geometry
SN - 0179-5376
ER -
ID: 369291347