Jacobi Ellipsoid

https://en.wikipedia.org/wiki/Jacobi_ellipsoid

34•perihelions•7mo ago

Comments

masfuerte•7mo ago

This is surprising, but less surprising when you realise it is rotating about one of the foci, not the centre.

perihelions•7mo ago

Hmm, that can't be true; it's of uniform density, and the thing it's rotating about has to be its center of mass.

What's odd about it (to me) is the optimal solution isn't symmetric (cylindrically symmetric). It's an intuition trap that you'd expect symmetric solutions. If the Wikipedia history is to be believed, Lagrange fell for this wrong assumption, and there was a 45-year gap before anyone noticed the subtle wrongness of it.

namibj•7mo ago

Remember the spontaneously flipping spinning hammer on the ISS?

...rotation for sufficiently non-spherical(-cow) objects in free fall is kinda weird.

kgwgk•7mo ago

The ellipsoid can rotate around its minor axis (more stable), around its major axis (also stable) or around its medium axis (unstable, the direction of the axis will “flip”). The three axes go through the center.

adrian_b•7mo ago

It has a lower symmetry than cylindrical, but it still has axes of symmetry and planes of symmetry and a center of symmetry.

The equilibrium condition only forces the symmetry condition that the points of intersection between any straight line that passes through the center of rotation with the surface of the body must be equidistant to the center of rotation.

This symmetry condition is satisfied by the symmetry group of the rectangular parallelepiped, which is also the symmetry group of the triaxial ellipsoid.

MycroftJones•7mo ago

That page didn't have a formula, in either cartesian or polar coordinates, for the shape of the object. Lots of formulas, but I didn't see anything I could use to create a 3d mesh and print one of these things out on my printer.

groos•7mo ago

It's the general ellipsoid formula: x*2/a*2 + y*2/b*2 + z*2/c*2 = 1, where a, b and c are all unequal. The interesting part is actually that this shape could be made of a liquid, held by gravitation and maintain this asymmetrical shape. Normally, one would imagine it would be ellipsoid of revolution, where two of the axes are equal.

https://en.wikipedia.org/wiki/Ellipsoid

To make a plot in software such as JMol, you really need parametric equations, which are also given in the above page.

dabluecaboose•7mo ago

Relevant to this is the J2 Perturbation [1], commonly used when accurately modeling the Earth as an oblate spheroid (like in the article image, but less drastic) rather than a perfect sphere. This has resultant effects on orbits, such as the "gravity wells" in the GEO belt at 105degW and 75degE. There are higher-order perturbations [2] as you closer approximate the Earth's actual shape, such as J3, J4, etc.

[1] https://ai-solutions.com/_freeflyeruniversityguide/j2_pertur...

[2] https://oer.pressbooks.pub/lynnanegeorge/chapter/chapter-10-...

Al Lowe on model trains, funny deaths and working with Disney

Hoot: Scheme on WebAssembly

Stories from 25 Years of Software Development

OpenCiv3: Open-source, cross-platform reimagining of Civilization III

Reinforcement Learning from Human Feedback

The AI boom is causing shortages everywhere else

The Waymo World Model

Start all of your commands with a comma (2009)

Selection Rather Than Prediction

Vocal Guide – belt sing without killing yourself

Speed up responses with fast mode

France's homegrown open source online office suite

U.S. Jobs Disappear at Fastest January Pace Since Great Recession

Coding agents have replaced every framework I used

A Fresh Look at IBM 3270 Information Display System

72M Points of Interest

Software factories and the agentic moment

Unseen Footage of Atari Battlezone Arcade Cabinet Production

Where did all the starships go?

Show HN: Kappal – CLI to Run Docker Compose YML on Kubernetes for Local Dev

Show HN: Look Ma, No Linux: Shell, App Installer, Vi, Cc on ESP32-S3 / BreezyBox

Learning from context is harder than we thought

Monty: A minimal, secure Python interpreter written in Rust for use by AI

Making geo joins faster with H3 indexes

Hackers (1995) Animated Experience

Sheldon Brown's Bicycle Technical Info

Ga68, a GNU Algol 68 Compiler

Show HN: If you lose your memory, how to regain access to your computer?

An Update on Heroku

Show HN: I spent 4 years building a UI design tool with only the features I use