Bézier by hand

Four control points, one smooth curve. Drag any point. Switch on the construction lines and slide t to watch de Casteljau's method fold the control polygon down to a single point on the curve — averaging, then averaging the averages.

The maths: a point on the curve is (1−t)³P₀ + 3(1−t)²t P₁ + 3(1−t)t² P₂ + t³P₃ — a weighted average whose weights always sum to 1. Read Bézier curves are just weighted averages.