Triangle Inequality Average colouring for escape time fractals:

t = \sum \frac{ |z| - |(|z-c|-|c|)| }{ |(|z-c|+|c|)| - |(|z-c|-|c|)| }
where $c$ is constant (determined by pixel) and $z$ varies (by iteration of the fractal formula)

This suffers from catastrophic cancellation when $|z| << |c|$, but it's possible to rearrange the calculation. Let $z=x+iy$ and $c=a+ib$, then

|z-c|-|c| = \frac{ x^2 + y^2 - 2ax - 2by }{ \sqrt{ |z-c|^2 + |c|^2 } + |c| }

Now my deep zooms with TIA colouring no longer have glitches around mini-sets due to loss of precision.

Still need to add some tweaks (like only considering the last few iterations before escape) to make it look good, with deep zooms the contrast just washes right out with standard TIA.

Somewhat more interesting than deep zooming is using it for interior regions with a relatively low iteration count.


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