Bézier Curves

Numerical Analysis · Axiom Academy

The curves that power computer graphics Bézier curves are parametric curves defined by control points . Unlike interpolating polynomials, Bézier curves don't pass through all control points — instead, the points define the curve's shape. Drag the control points to reshape the curve! Developed by Pierre Bézier (Renault) and Paul de Casteljau (Citroën) in the 1960s for car design. Now fundamental to fonts, vector graphics, and animation. A Bézier curve of degree n with control points P₀, P₁, ..., Pₙ is: Where Bᵢₙ(t) are the Bernstein polynomials : The most common case uses 4 control points: An elegant recursive way to evaluate Bézier curves: repeatedly find midpoints! For t = 0.5, we compute midpoints at each level until we reach the point on the curve. This is numerically stable and geometrically intuitive! B(0) = P₀ and B(1) = Pₙ — the curve passes through the first and last control points. B'(0) is parallel to P₀P₁ and B'(1) is parallel to Pₙ₋₁Pₙ. The curve lies entirely within the convex hull of its control points. Transforming control points transforms the curve identically. ∑Bᵢₙ(t) = 1 for all t ∈ [0,1] — coefficients sum to 1. Reversing control points reverses the curve direction. Every letter you see is made of Bézier curves! Adobe Illustrator, Inkscape, and web SVG use cubic Béziers. Motion along smooth curves, easing functions. Car bodies, airplane wings, product design. A letter 'S' made from cubic Bézier curves

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