TL;DR
This paper investigates the geodesic complexity of a cube, establishing that it is exactly two greater than its topological complexity through analysis of cut loci.
Contribution
It provides a precise comparison between geodesic and topological complexity for a cube, a novel result in geometric topology.
Findings
Geodesic complexity of a cube exceeds topological complexity by 2
Analysis of cut loci is key to understanding geodesic complexity
The result clarifies the relationship between shortest path rules and topological structure
Abstract
The topological (resp. geodesic) complexity of a topological (resp. metric) space is roughly the smallest number of continuous rules required to choose paths (resp. shortest paths) between any points of the space. We prove that the geodesic complexity of a cube exceeds its topological complexity by exactly 2. The proof involves a careful analysis of cut loci of the cube.
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Taxonomy
TopicsDigital Image Processing Techniques · Computational Geometry and Mesh Generation · Advanced Graph Theory Research