Distance and slope: shading with a derivative

Escape time tells you when an orbit left; it says nothing about the geometry it left behind. But the renderer can also track how fast the orbit responds to a nudge in $c$ — its derivative. Differentiate the rule $z_{n+1} = z_n^2 + c$ with respect to $c$ and the chain rule hands you a second recurrence to run alongside the first:

$$z'_{n+1} = 2\,z_n\,z'_n + 1, \qquad z'_0 = 0$$

(in the Julia plane, differentiate with respect to $z_0$ instead: the same recurrence without the $+1$, seeded $z'_0 = 1$). One extra multiply-add per step — almost free — and two more ways of seeing open up. Same view, three treatments:

Smooth colouring. The baseline: exterior coloured by fractional escape count alone. c ≈ −0.743643887 + 0.1318259042i · span 1.5×10⁻³ · open this view in Mandala →

Smooth colouring is the baseline from the escape-time chapter: palette by fractional escape count, nothing else.

Distance shading. The orbit’s derivative estimates how far each pixel is from the set; darkening by that distance inks the boundary like an etching. c ≈ −0.743643887 + 0.1318259042i · span 1.5×10⁻³ · open this view in Mandala →

Distance estimation combines the orbit's final size with that derivative into an estimate of the distance from the pixel to the set itself:

$$d \;\approx\; \frac{\lvert z_n \rvert \, \ln \lvert z_n \rvert}{\lvert z'_n \rvert}.$$

The intuition: $\ln|z_n|$ measures how far along the escape the orbit is, and dividing by $|z'_n|$ converts that to distance in the $c$-plane — where the derivative is huge, a tiny step in $c$ swings the orbit wildly, so the boundary must be close. Classical function theory (the Koebe distortion theorem) makes this honest: the estimate is correct up to a small constant factor — near the set it never lies by more than about 4×, so darkening pixels by $d$ inks the boundary like an etching. Filaments far too thin for any escape-time palette to resolve become crisp lines. This is the mode that shows you where the set actually is.

Its limitation is inherited from the families chapter: the formula needs a complex derivative. For the non-analytic families (Burning Ship, Tricorn, Celtic, Phoenix), Mandala carries the analytic degree factor as an approximation for a useful boundary-highlight aesthetic; it does not pretend that the conformal-distance guarantee survives the folds.

Slope shading. Treat escape height as terrain and light it from an angle: the same derivative becomes a surface normal. c ≈ −0.743643887 + 0.1318259042i · span 1.5×10⁻³ · open this view in Mandala →

Slope shading reads the same derivative as terrain. Treat the smooth escape count as a height field over the plane; its gradient — assembled from $z_n$ and $z'_n$, no extra iteration needed — gives a surface normal, and lighting that normal from a chosen angle turns the exterior into relief, like a raking-light photograph of a carved surface.

In the implementation the complex direction $u=z_n/z'_n$ is normalised to $\hat n$, then a Lambert-like term controls brightness:

$$s=\operatorname{clamp}\!\left( \frac{\hat n\cdot L+h}{1+h},0,1\right).$$

$L$ is the chosen light direction and $h$ is the ambient-height term: raising $h$ flattens the relief; lowering it deepens the shadow side.

Details that keep it honest

The interior stays solid black in every mode — Mandala never invents structure where the mathematics says "bounded". Distance shading converts $d$ from complex units to screen pixels before applying its ramp, so a one-pixel line remains visually one pixel wide as you zoom. Width, falloff and an invert option control whether the boundary becomes dark ink or a bright glowing line. Slope exposes light angle and relief depth.

Both modes run on the GPU, $f64$ CPU and double-double CPU tiers. At the extreme perturbation tier Mandala currently does not perturb the derivative; it passes a zero derivative and both modes degrade safely to ordinary smooth colouring. Newton and Nova also bypass exterior shading because their basin/ hybrid colour paths are not escape-distance fields. Exports preserve whichever supported treatment is actually visible on screen.