Closed curves in n-dimensional discrete space

Authors:
Grit Thürmer
Affiliations:
Computer Graphics, Visualization, Man-Machine Communication Group, Faculty of Media, Bauhaus-University Weimar, 99421 Weimar, Germany
Venue:
Graphical Models - Special issue: Discrete topology and geometry for image and object representation
Year:
2003

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Abstract

A new definition of closed curves in n-dimensional discrete space is proposed. This definition can be viewed as a generalization of closed quasi curves and is intended to overcome the limitations of the known definitions for practical purposes. Following the proposed definition, a set of points forms a closed curve in discrete space if the set admits a parameterization, i.e., there exists a Hamiltonian cycle in the set. Adjacencies that do not indicate the parameterization are allowed only between points that are "close to each other" along the parameterization. Additionally, it is proven that discrete curves satisfying the new definition in two-dimensional discrete space have the Jordan property.