DeepAI AI Chat
Log In Sign Up

Topological Art in Simple Galleries

by   Daniel Bertschinger, et al.

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.


page 1

page 2

page 3

page 4


On sets of n points in general position that determine lines that can be pierced by n points

Let P be a set of n points in general position in the plane. Let R be a ...

Topological Universality of the Art Gallery Problem

We prove that any compact semi-algebraic set is homeomorphic to the solu...

A new compressed cover tree for k-nearest neighbour search and the stable-under-noise mergegram of a point cloud

This thesis consists of two topics related to computational geometry and...

Complexity of Shift Spaces on Semigroups

Let G=〈 S|R_A〉 be a semigroup with generating set S and equivalences R...

Favourite distances in 3-space

Let S be a set of n points in Euclidean 3-space. Assign to each x∈ S a d...

Towards Stratification Learning through Homology Inference

A topological approach to stratification learning is developed for point...

Nonparametric Topological Layers in Neural Networks

Various topological techniques and tools have been applied to neural net...