claudeGuillaume Collombet (with assistance from LLM...) on fractal chats discord suggested using #MoebiusTransformation instead of #LinearApproximation to accelerate #MandelbrotSet #DeepZoom .
The advantage is that the quadratic term of the perturbation formula is now exact, with a cubic remainder term. for single step, assuming that z < sqrt(eps) 2Z makes this remainder be lost in rounding noise, while for the linear approximation the corresponding condition is z < eps 2Z.
given two moebius with validity radii, the validity radius for their composition can presumably be calculated without much difficulty.
eps is typically 1e-16 to 1e-8, depending on how much precision loss you think you can get away with.
Guillaume claims moebius gives a practical computational efficiency improvement over linear. I haven't tried it myself yet.
However, while I think it should work well for #JuliaSet, where composition of two moebius in z gives another moebius in z, the nonzero c term for mandelbrot set ends up with something with zc and c^2 terms when composing the two functions, which Guillaume handwaves away reasoning that c is usually tiny and z is also small most of the time. I'm not 100% convinced that iteration (composition coefficients getting larger) wont eventually make them significant...
Jeroen van Dorp
Microfractal 🫖
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