Sphere eversion with a ruled surface.
In the intermediate stage, the surface is very similar to half-way model with a Boy's surface.
Reference:
Analytic sphere eversion using ruled surfaces
https://arxiv.org/abs/1711.10466
More about the sphere eversion problem and the method called "Thurston's corrugations":
Outside In: https://www.youtube.com/watch?v=sKqt6e7EcCs
More about Boy's surface:
https://www.youtube.com/watch?v=wK1-thAV0PE
Another method, with many variations including Boy's surface:
The Optiverse: https://www.youtube.com/watch?v=cdMLLmlS4Dc
And an interactive demo: https://rreusser.github.io/explorations/sphere-eversion/
