Visual Tools for Hyperbolic Geometry

A guided research project in Geometry & Dynamics.

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Experience Research Geometry & Dynamics

Question

How can we make the unfamiliar geometry of hyperbolic space visible and interactive?

A visual model turns abstract hyperbolic rules into objects that can be moved and tested.
A visual model turns abstract hyperbolic rules into objects that can be moved and tested.
A hyperbolic tiling drawn inside a Poincaré disk.
A Poincaré-disk rendering from the hyperbolic-geometry research repository.

Goals

  • Draw geodesics, isometries, and tilings in standard hyperbolic models.
  • Build a tool that lets users test geometric ideas by changing the picture.

Method

Convert hyperbolic objects into coordinates that can be drawn on a flat screen.

  • Choose the Poincaré disk or upper-half-plane model.
  • Compute geodesics and Möbius transformations from user-selected points.
  • Render the results and check that hyperbolic distances and angles behave correctly.