Escape-time rendering
Escape-time rendering asks how long an orbit takes to cross a proven bailout radius. It is simple enough for a first program and subtle enough that every numerical and visual choice matters.
Map, iterate, classify
Each pixel is mapped into the relevant complex plane. The renderer initializes an orbit, repeats the formula and stops when the magnitude exceeds its bailout or the maximum iteration count is reached. Escaped and unresolved points are then passed to coloring.
Each pixel first maps to a complex coordinate. The same iteration then runs until the orbit crosses the bailout radius or reaches the limit. This order should be verified without color: known interior points remain, distant points escape quickly, and boundary points respond predictably to greater depth. Only a correct classification is ready for visual mapping.
The iteration limit is not the set
A point that has not escaped after 500 iterations is not thereby proven inside for every formula. Raising the limit reveals slowly escaping regions and changes exterior bands. Interior tests and periodicity detection can improve confidence and speed.
The iteration cap is observation time, not the definition of the set. Raising it can reclassify late-escaping pixels while proven exterior pixels remain stable. An interactive preview may begin with a smaller cap but must continue after input stops. If it remains permanently coarse, a performance technique has become a rendering defect and may create false interior islands.
Smooth values remove integer bands
Raw escape iteration is an integer and creates visible stripes. A continuous escape estimate uses the final magnitude to interpolate between counts, producing smoother gradients. It changes the coloring coordinate, not the orbit classification.
For quadratic escape-time images, integer escape count can be interpolated using magnitude at escape. The familiar continuous quantity ν=n+1−log₂(log|zₙ|) removes hard integer bands without inventing additional orbits. It depends on formula and power. Applying one smoothing expression to every fractal can create seams or false phase.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Smooth Iteration Count for General PolynomialsLinas Vepštas
- The Science of Fractal ImagesSpringer
Continue the idea
The Mandelbrot set
What the Mandelbrot set contains, how z² + c creates its boundary and why deep zoom reveals related rather than identical copies.
Fractal coloring beyond iteration bands
A guide to smooth iteration, potential, distance, angle, orbit traps, interior coloring and perceptual gradients.
Distance estimation and contour structure
How derivative-aware estimates approximate distance to a fractal boundary and support contours, antialiasing and relief.