Next up: elliptic transformations



Next up: elliptic transformations

These are analogous to Euclidean rotations. The motion fixes an entire line in hyperbolic space, the core geodesic of the banana shape. Points traverse parallel circles on the banana, which is a cylinder in hyperbolic space.

For this animation, it would also have been natural to pick a geodesic plane orthogonal to the banana as a surface to render. Such planes are filled with concentric circles, culminating in an ideal circle (a member of the family of concentric circles we see here on the boundary).

Something to add to the shader. http://bit.ly/2BNdwS4

Comments

  1. John BaezFebruary 24, 2018 at 10:55 PM

    Excellent! Banana math! :-)

    ReplyDelete
  2. Roice NelsonFebruary 24, 2018 at 10:58 PM

    We humans are a bunch of monkeys after all! :D

    ReplyDelete
  3. Saul SchleimerFebruary 25, 2018 at 10:00 AM

    Next next up: parabolics?

    ReplyDelete
  4. Roice NelsonFebruary 25, 2018 at 10:15 AM

    Saul Schleimer, after hyperbolic :)

    An earlier, less colorful version of parabolics is here:

    plus.google.com - Discoball! ...or a shader that demonstrates a parabolic transformation in th...

    It would be cool to do an animation of the two fixed points of an elliptic or hyperbolic approaching each other and merging into the single fixed point of a parabolic, to show that a horosphere is weirdly also a limit of a hyperbolic cylinder of increasing radius.

    ReplyDelete

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