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There are a number of demonstrations of rendering in hyperbolic or spherical geometry (See here for example). Is there any way to do something similar such that light travels on the $L_\infty$ shortest path?

I'm asking, since I was doing some shortest path calculations with the $L_\infty$ metric, and they're very nonintuitive, so warping the space somehow to make them appear more intuitive would be very helpful.

I'm interested in both full on rendering, or warping a 2/3-D lattice such that $L_\infty$ shortest paths become $L_2$ shortest paths. If this is not possible, why is that the case?

Davis Yoshida
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  • Aren't straight lines $L_{\infty}$-shortest paths? – Bananach Nov 03 '21 at 18:41
  • @Bananach They can be, but to go from say (0,0) to (10, 0), both a straight line and the path (0, 0) -> (5, 5), -> (10, 0) have the same length, which makes eyeballing path lengths somewhat counterintuitive. – Davis Yoshida Nov 04 '21 at 01:34

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