Bézier Curves
Drag the control points to reshape the curve. The animated construction shows de Casteljau's algorithm: linear interpolation applied recursively until a single point traces the curve. Used everywhere from fonts to animation paths.
B(t) = Σ C(n,i) · tⁱ · (1−t)^(n−i) · Pᵢ | de Casteljau: Pᵢʳ(t) = (1−t)·Pᵢʳ⁻¹(t) + t·Pᵢ₊₁ʳ⁻¹(t)
Pierre Bézier (Renault, 1962) and Paul de Casteljau (Citroën, 1959) independently invented these curves for CAD systems. They are now the standard representation in PostScript, PDF, SVG, HTML Canvas, and TrueType/OpenType fonts.
De Casteljau's algorithm: to evaluate a Bézier curve of degree n at parameter t, repeatedly interpolate between adjacent control points with ratio (1-t):t. This takes n rounds and produces the point on the curve.
Key properties: the curve lies within the convex hull of its control points (convex hull property); the curve interpolates the first and last control points; and a subdivided Bézier curve is two Bézier curves, making subdivision efficient.