Spiderwebs on the Sphere and an Isoperimetric Theorem

TL;DR

The work addresses rigidity of spherical tensegrities arising from projecting convex inscribed polytopes onto the unit sphere, and links this to spherical isoperimetric results. By restricting to inscribed polytopes with face circumcenters inside or on their faces and treating edges as geodesic cables, it proves a semi-global rigidity result via isoperimetric optimization on $\mathbb{S}^2$. A central contribution is linking extrema of the enclosed area, under fixed edge lengths, to uniqueness up to rotation, yielding rigidity for the spiderweb-like configurations. The results illuminate pre-stress stability and suggest mechanisms to recover polytopal realizations from rigid spherical tensegrities.

Abstract

Here we present a rigidity result in a global (semi-global, homotopy) setting for a restrictive class of polytopes, those that can be inscribed in a unit sphere, with some additional conditions. The proof of the rigidity result for cabled frameworks on the surface of the sphere uses classical isoperimetric ideas.

Figures (10)

• Figure 2.1: This is a stereographic projection into the plane of the points and line segments defined in the proof of Lemma \ref{['hemi']} and Lemma \ref{['lemma:closed-hemi']}. Circles in $\mathbb{S}^2$ are projected into circles and lines in the plane. The south pole (the point at infinity) is the only point in the sphere not projected to a point in the plane.
• Figure 2.2: The green circular segments, the regions between the spherical geodesic segments and the bounding circle, are attached rigidly to the spherical polygon. This is a picture of the analogous situation in the plane.
• Figure 3.1: A Queen Dido polygon: If any single edge, but not the longest, is increased/decreased the area is increased/decreased. When the longest edge of the polygon is changed, fixing the rest, the area of the polygon is decreased.
• Figure 3.2: This shows the stereographic projection of the level lines of the area of a geodesic triangle on the sphere with one fixed side of length $a$. They are circles, each of which intersect the antipodes of the end points of the geodesic line segment of length $a$.
• Figure 3.3: This shows that when a triangle has its circumcenter in its interior, then the length of the edge on the boundary of its containing polygon (on the top of the polygon here) determines how the area of the whole polygon increases or decreases. The colored polygons have a fixed area in the two figures. When the circumcenter of a polygon lies inside the polygon, there always is a triangle, as in the figure, where decreasing the outer edge will decreas the area.

Theorems & Definitions (19)

• Lemma 1
• proof
• Lemma 2
• Theorem 2.1
• Lemma 3
• proof
• Theorem 3.1
• Corollary 3.2
• proof
• Theorem 4.1