# Basics of geodesic fiber winding vs isotensoids in pressure vessel construction (with focus on future habitats in space)

### Background:

TechCrunch's July 27, 2024 Max Space reinvents expandable habitats with a 17th-century twist, launching in 2026 references isotensoids several times.

I can find mention of isotensoid in the following Wikipedia pages:

but I still don't have a good clear idea what an isotensoid actually is.

Searching lead me to Guo et al. (2020) Design of winding pattern of filament-wound composite pressure vessel with unequal openings based on non-geodesics where the introduction nicely but briefly introduces geodesic winding patterns but only to discuss their limitations and all the newer non-geodesic winding patterns and algorithms.

To broaden the designability, the non-geodesics winding is more and more used in the filament winding on complex parts. By employing the friction between the filament bundles and its supporting surface, the non-geodesics significantly enlarge the available winding path.

So far, I have found that in mathematics, a geodesic of a surface is not necessarily the shortest path between a pair of given points A and B. It seems to be more of a natural path or some kind of trajectory that can be found with an ordinary differential equation (ODE) solver, perhaps in some way similar to how we might integrate the trajectory of a spacecraft.

Wikipedia's Geodesic; introduction; examples includes the following image, which seems to be nothing to do with finding the shortest path. I have a hunch that it's actually a closed path but I can't tell.

Source click for larger

Transpolar geodesic on a triaxial ellipsoid, case A. Vital statistics: a:b:c = 1.01:1:0.8, β1 = 90°, ω1 = 39.9°, α1 = 180°, s12/b ∈ [−232.7, 232.7], orthographic projection from φ = 40°, λ = 30°. The geodesic is found by solving the ordinary differential equations for the free motion of a particle constrained to the surface of the ellipsoid; the solution is carried out in Cartesian coordinates.

### Question:

Isotensoids and geodesics both seem to be mathematical concepts and tools related to construction of pressure vessels, including those for spaceflight. While geodesic fiber winding seems to be passé, the TechCrunch article suggests that there will be isotensoid pressure vessels in Earth orbit in a few years, with one long-term development target being habitable structures for people.

Is it possible to explain the basics of geodesic fiber winding vs isotensoids in pressure vessel construction in the form of a nice Stack Exchange answer in such a way that I'd be able to use each in a sentence and not sound confused?

For the geodesic trajectory what does "the ordinary differential equations for the free motion of a particle constrained to the surface" actually mean? For example, if I make a hollow ellipsoid in zero gravity and put a tiny ball bearing inside it and give it a kick - is its trajectory that of a geodesic?