Procedural generation and noise

Level UniversityDifficulty ★★★★★Application⌖ Open in the map

What is it?

Generating terrain, clouds and textures from smooth random functions (Perlin noise) summed over octaves (fractal Brownian motion). Smooth interpolants ( $6t^5 - 15t^4 + 10t^3$ , with zero first and second derivatives at the ends) avoid visible grid artefacts; analytic derivatives give normals for lighting.

Formulas

\operatorname{fBm}(x) = \sum_{k=0}^{K-1} a^k\,\operatorname{noise}(2^k x), \quad a \approx \tfrac12

The mathematics behind it

Fractals★★★★★frequent
Fractal noise (fBm: octaves of Perlin noise) generates terrain, clouds and textures in games and films.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.