This paper's Theorem 1.8 seems to extend the Koebe 1/4 Theorem of #ComplexAnalysis into n-dimensional real spaces, which could be #VeryUseful in getting reliable¹ #DistanceEstimate formulas for 3D #fractals.

> Quasiconformal analogues of theorems of Koebe and Hardy-Littlewood.
> K. Astala and F. W. Gehring
> Michigan Math. J. Volume 32, Issue 1 (1985), 99-107.
> https://projecteuclid.org/euclid.mmj/1029003136

Has some pre-requisites I need to research further, like knowing what K-quasiconformal means. If only I understood it enough to calculate the coefficient c (which is 4 for conformal complex functions), which depends only on the K of the function and the dimension of the space...

The "integrate the log of the Jacobian over the largest ball that fits inside the domain" part might be tricky in practice too, maybe some luck will mean something turns out to be harmonic so it can be evaluated at the center only? Not sure about any of this. Maybe I'm in over my head...

#AmReading #maths

¹ reliable means "this is a proven lower bound" so that sphere-marching renderers will never overstep