This is an interactive visualization of our algorithm for mapping conformally the hyperbolic plane to arbitrary shapes. It helps to understand the main ideas of our paper here. The implementation of our algorithm, 'newconformist', is available here. The algorithm used in this visualization is based on the same main ideas as newconformist, i.e., it maps the shape to the band model.

• Click on the large squares (pixels) close to the edges of the shape to the left to change the shape. The algorithm will map the hyperbolic plane conformally to the desired shape.
  
  
• You can view only the original shape, view the end result, or see the construction using the band model.
  
  
• The system of equations is solved iteratively, to make it react immediately to changes (which looks cool). You can reset the iteration, solve the system of equations quickly using Gaussian elimination, or visualize the difference bewteen the slow and fast elimination order.
  
  
• For a clear presentation there is just about a hundred pixels. Of course this reduces the quality of the result, but you can double the resolution.
  
  
• Some predesigned shapes: square, rectangle, T-shape, and snake.
  
  
• You can also use a different pattern.
  
  
• If you go fullscreen, press 'v' for a menu.
  
  
• Only the basic algorithm, no extensions for branching shapes and annuli.
  
  
• It uses the period of the tessellation to draw it nicely everywhere along the main line, but it does not know about vertical periods, so branches will be drawn less nicely (in fact it draws the tessellation in the band model and then uses that tessellation as a texture).

Play HyperRogue online to get intuitions about hyperbolic geometry. Look here for more visualizations.



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