Uploaded August 2020 | Updated September 2026, 1 week ago

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PSL(2,R) is a Lie group, which can be seen as the space of isometries of the hyperbolic plane. It has its usual metric, but we can also define the metric tensor g at one point, and use the Lie group to define a different homogeneous geometry.

Here, we are using g(v,w) = ⟨Av,Aw⟩ where A(x,y,z) = (0.4*x,0.5*y,z), and ⟨⟩ is the usual inner product. The image in the corner shows the isometry of the hyperbolic plane that the current camera position corresponds to. The "tentacles" corresponds to the heptagons in the H^2 scene.

In the last seconds we show the transition from the standard metric to the weirdly stretched one.

Just like in the spherical case, mostly to see what it would look like -- it seems difficult to understand what is going on here (other than what was said in the last paragraph), or to find any practical uses. But it looks quite fun!

Spherical analog: youtu.be/VJyv5kfQbIA
~SL(2,R) but stretched along the fiber -- much more tame: youtu.be/Hg-IW6XfgZY or youtu.be/YUwbFLFBSCk
Standard PSL(2,R): youtu.be/y0dvajkAHlA

Made with the HyperRogue engine!