Minimum Number of Spherical Geodesics in Disjoint Arrangements

Core Concepts

Any k-oriented spherical diagram (SD) or spherical occlusion diagram (SOD) has a minimum number of arcs and swirls, depending on the configuration of the poles.

Abstract

The content discusses the study of arrangements of geodesic arcs on a sphere, where all arcs are internally disjoint and each arc has its endpoints located within the interior of other arcs. The author establishes fundamental results concerning the minimum number of arcs in such arrangements, depending on local geometric constraints such as one-sidedness and k-orientation.
The key highlights and insights are:

The author generalizes and settles an open problem, proving that any such arrangement has at least two clockwise swirls and at least two counterclockwise swirls.

For non-degenerate k-oriented SDs and SODs, the author provides the minimum number of arcs and swirls for all possible configurations of the poles.

For degenerate k-oriented SDs and SODs with k ≤ 5, the author also determines the minimum number of arcs and swirls for all possible configurations of the poles.

The author shows that for nearly all of the alignment configurations, there are SDs and SODs that simultaneously minimize the number of arcs and the number of swirls.

The author introduces the concept of attractor hulls and sliding walks, which are used to derive the main results.

Stats

There are no key metrics or important figures used to support the author's key logics.

Quotes

"Any SD has at least two clockwise swirls and two counterclockwise swirls."
"For nearly all of the alignment configurations listed in Table 1, there are SDs and SODs that simultaneously minimize the number of arcs and the number of swirls."

Key Insights Distilled From

Minimal Arrangements of Spherical Geodesics

by Giovanni Vig... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2311.03255.pdf

Deeper Inquiries

How can the results be extended to higher-dimensional geometric arrangements beyond spheres

The results obtained for minimal arrangements of spherical geodesics can be extended to higher-dimensional geometric arrangements beyond spheres by considering the analogous concepts in higher dimensions. For instance, in 4-dimensional space, one could explore arrangements of geodesic hypersurfaces on a 3-sphere. By generalizing the definitions and properties established for spherical geodesics to higher dimensions, it is possible to investigate the minimum number of hypersurfaces, their intersections, and the combinatorial structures that arise in such arrangements.

What are the implications of these findings for practical applications in areas such as computer graphics, robotics, or architectural design

The findings from the study of minimal arrangements of spherical geodesics have significant implications for practical applications in various fields. In computer graphics, understanding the minimum number of geodesics required to cover a sphere can lead to more efficient algorithms for rendering and modeling spherical objects. In robotics, the concept of minimal arrangements can be applied to path planning and motion control in spherical environments. For architectural design, the results can inform the layout of structures on curved surfaces, optimizing visibility and spatial organization.

Is there a deeper connection between the combinatorics of these spherical arrangements and other areas of mathematics, such as graph theory or topology

There is a deep connection between the combinatorics of spherical arrangements and other areas of mathematics, such as graph theory and topology. The swirl graphs and visibility maps studied in the context of spherical geodesics can be viewed as special cases of graph representations of geometric structures. The minimum number of arcs in arrangements of geodesics corresponds to extremal graph theory problems, while the topological properties of these arrangements relate to concepts in geometric topology. This intersection of geometry, graph theory, and topology provides a rich area for further exploration and interdisciplinary research.