A Bezier curve of degree n is determined by control points P0,P1,…,Pn. The de Casteljau algorithm evaluates it at parameter t∈[0,1] by repeated linear interpolation: set Pi(0)=Pi, then for each level k=1,…,n compute Pi(k)(t)=(1−t)Pi(k−1)+tPi+1(k−1),i=0,…,n−k.
After n reductions, a single point P0(n)(t)=B(t) remains — that is the Bezier curve at t. Equivalently, B(t)=∑i=0n(in)(1−t)n−itiPi, the Bernstein form, but de Casteljau is the numerically stable, geometric way to see it: every interior point is just a weighted average. Here n=5 (six control points). The parameter t sweeps from 0 to 1 over four seconds, holds, and restarts. Gray segments fanning down the levels show the intermediate interpolants; the orange dot at the bottom of the fan is B(t), and its trajectory traces the blue curve. Drag a control point to reshape the curve. Inspired by Gabriel Peyré's classical-geometry visualizations.