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JuliaManifolds / Manifolds.jl

Manifolds.jl provides a library of manifolds aiming for an easy-to-use and fast implementation.
https://juliamanifolds.github.io/Manifolds.jl
MIT License
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Add projection for the hyperbolic space #79

Open mateuszbaran opened 4 years ago

mateuszbaran commented 4 years ago

As discussed here: https://github.com/JuliaNLSolvers/Manifolds.jl/pull/71#issuecomment-569412815

kellertuer commented 4 years ago

I would still like to see a reference /literature for such a projection or a verification that your proposed procedure is the closest point on the manifold to said point from the embedding.

mateuszbaran commented 4 years ago

Right, I'd like to verify it too. I tried to derive it, it looks deceptively simple since there are very nice symmetries but the Minkowski distance is so unintuitive 🙁 .

kellertuer commented 4 years ago

...and intuitively, I even think your formula is right ;)

kellertuer commented 4 years ago

I checked the reference again (see https://math.stackexchange.com/questions/3175081/projection-of-point-onto-closest-point-on-geodesic-in-hyperbolic-geometry-hyper/3176477#3176477) and it refers to projection a point on the hyperboloid model onto a geodesic on the hyperboloid model, so its not the project(M,x)for a point from the embedding we are looking for. Actually there is a thread ( https://math.stackexchange.com/questions/3218487/orthogonal-projection-of-point-in-ambient-space-onto-hyperboloid ), so I am not sure whether there exists a formula this issue is looking for?