Physics / Math
Distance-field fractals
Surfaces with no mesh: a distance estimator and sphere tracing draw the whole thing.
What happens
None of these surfaces has a mesh or an equation for its points. What exists is a distance estimator, a function answering how far the nearest surface is from here, and sphere tracing walks each ray by exactly that distance until it arrives. The estimate is a lower bound, never a measurement, which is why marching by it can never step through the surface. The bound comes from the escape-time derivative: the orbit is iterated together with the derivative of that iteration. Raise the iteration count and the bound tightens, revealing finer structure at the cost of steps, and the panel prints that cost.
Equations used in the module
Equations distance estimator: d(p) ≈ |z| ln|z| / (2 |z′|) sphere tracing: p ← p + d(p) · ray Mandelbulb: z ← z^n + c in spherical coordinates
