Live demo: click here

Description:

The pseudosphere is a surface of constant negative curvature. Also shown are three other representations of hyperbolic geometry: the upper half-plane model, the Poincaré disk model and the Klein disk model.

The blue and green lines are geodesics ("straight lines") in the four models. You can drag their end points to move them around (except on the 3D view).

To map the upper half-plane onto the pseudosphere, everything above the red line (y=1) is wrapped around the funnel, like a rolled-up sheet of paper. The orange lines in each view map onto the single orange line on the pseudosphere. A geodesic that crosses multiple orange lines will therefore wind itself more than once around the pseudosphere.

A geodesic on the pseudosphere is the path an ant would take if it were to walk straight forwards. Remarkably, the complicated paths created always unwrap to form straight lines in the Klein disk model. There's a deep connection here with general relativity, where inertial objects may take complicated paths in space but are actually moving along straight lines in spacetime. See here for a similar web app that visualises how this works.

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Thanks to Rickard Jonsson for helping me understand how geodesics work. Thanks also to Tadao Ito for making these amazing pages explaining how the pseudosphere maps onto horocycles in the Poincaré disk model: [1] [2]