Here's an atypical conformal model of the hyperbolic plane.



Here's an atypical conformal model of the hyperbolic plane. The Poincaré disk is mapped to the entire plane via the Joukowsky transformation (https://www.johndcook.com/blog/2016/01/31/joukowsky-transformation/), compressing the boundary-at-infinity to the interval [-1,1] on the real axis.

Has this model appeared in the literature? If so, what is it named?

Comments

  1. Melinda GreenOctober 7, 2018 at 8:28 PM

    Neat. Rotated 90 degrees clockwise resembles a Buddhabrot.

    ReplyDelete
  2. Dmitry ShintyakovOctober 8, 2018 at 4:16 AM

    Here are a lot of other models, some of which are quite surprising:
    bulatov.org - Conformal Models of Hyperbolic Geometry

    ReplyDelete
  3. Roice NelsonOctober 8, 2018 at 11:31 AM

    Interesting observation Melinda Green. These may be related since the Joukowsky transformation I used was (z^2+1)/2z and the numerator is the transformation used to make a Mandelbrot.

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  4. Melinda GreenOctober 8, 2018 at 11:35 PM

    Roice Nelson I don't understand that, but is there anything you might do to increase the correspondence?

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  5. Roice NelsonOctober 10, 2018 at 3:40 PM

    Melinda Green, I'm not sure. Here it is rotated and not cut off, at least. Thinking a little more and pulling up the Mandelbrot wiki page, my previous thought doesn't really apply anyway.
    https://lh3.googleusercontent.com/yQfOYH4A3bh73-xARGYe2KuWrEXI_WU-i_S9H8u3CjWFGz17YvKa92Ay6XpqR7M_TJ_2V0tXHPXGTYiu9dRkdqNSmPf61u-JwKzM=s0

    ReplyDelete
  6. Melinda GreenOctober 11, 2018 at 3:05 PM

    Roice Nelson Now it looks like the Stay-Puff marshmallow man. :-)

    ReplyDelete
  7. Matthew ArcusNovember 25, 2018 at 11:30 AM

    Hi, this is interesting. I've just added the Joukovsky transform to this shader (hit 'j' to enable): shadertoy.com - Shadertoy

    ReplyDelete
  8. Roice NelsonNovember 25, 2018 at 7:16 PM

    Matthew Arcus, very cool! Thanks for sharing.

    ReplyDelete

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