Minimal Arrangements of Spherical Geodesics

TL;DR

This work fully characterizes minimal spherical arc arrangements under orientation constraints by introducing and analyzing SDs and SODs on the sphere. Using sliding-walk techniques and attractor-hull theory, the authors derive tight lower bounds for the number of arcs and swirls for all non-degenerate $k$-oriented diagrams, providing complete minima for $k=3,4,5$ and a canonical construction for $k=6$ that achieves the theoretical lower bounds. They also resolve an open problem by proving that every SD has at least two clockwise swirls and two counterclockwise swirls, extending the result to all SDs beyond the SOD case. The results link spherical visibility maps to extremal graph drawings and offer a foundation for understanding vertex-hidden visibility in restricted-orientation polyhedral environments, with extensive implications for geometric combinatorics and related visibility problems.

Abstract

We study 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. We establish fundamental results concerning the minimum number of arcs in such arrangements, depending on local geometric constraints such as "one-sidedness" and "k-orientation". En route to these results, we generalize and settle an open problem from CCCG 2022, proving that any such arrangement has at least two "clockwise swirls" and at least two "counterclockwise swirls".

Figures (13)

• Figure 1: A polygonal scene of six rectangles (left), where the central point $v$ is vertex-hidden. The portion of each edge that is visible to $v$ is radially projected onto a sphere centered at $v$, creating a geodesic arc. The resulting visibility map is an SOD (right).
• Figure 2: (a) An orthogonal polyhedron. (b) A $7$-edge-oriented polyhedron having only vertical and horizontal edges (and faces). (c) A $4$-face-oriented and $6$-edge-oriented polyhedron.
• Figure 3: (a) The area in yellow is the right-side region of a sliding walk. The eye of a clockwise swirl (in purple) can be found within this region by doing a right-handed sliding walk from its boundary. (b) The initial steps of a clockwise walk with fulcrum $p$ (in red).
• Figure 4: (a) The swirl in yellow shares two arcs with the contiguous swirl in green and two arcs with the non-contiguous swirl in purple. (b) An attractor hull and a sliding walk within its interior.
• Figure 5: (a) A $3$-oriented SD with exactly eight swirls and $12$ arcs. (b) A non-degenerate $4$-oriented SOD with exactly six swirls and ten arcs. Each arc has the same color as its vanishing points. Dashed lines mark the boundaries of octants and sextants, respectively.

Theorems & Definitions (66)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 6
• Theorem 7
• Proposition 8
• Proposition 9
• Proposition 10
• Definition 11