Bézier Curve — de Casteljau

About: The de Casteljau algorithm (Paul de Casteljau, Citroën, 1959) computes a Bézier curve by repeated linear interpolation. At parameter t, interpolate between each pair of control points to get n−1 new points, then repeat n times until one point remains — that is B(t). This is numerically stable and reveals the geometric structure: the curve lies within the convex hull of control points, and the tangent at t=0 is P₁−P₀, at t=1 is Pₙ−Pₙ₋₁. The equivalent Bernstein polynomial form B(t)=Σ C(n,k)t^k(1-t)^(n-k)Pₖ shows that Bézier curves are affinely invariant and variation-diminishing.