Bézier

A Bézier curve is a parametric curve frequently used in computer graphics, defined by a set of control points. The curves are named after Pierre Bézier, who popularized them in the 1960s for automotive design; similar ideas were developed by Paul de Casteljau around the same time.

A Bézier curve of degree n is defined by control points P0, ..., Pn. The curve is B(t) = sum_{i=0}^n B_{n,i}(t) P_i for t in [0,1], where B_{n,i}(t) = binomial(n,i) (1-t)^{n-i} t^i are Bernstein basis polynomials. Equivalently, the De Casteljau algorithm provides a numerically stable evaluation via repeated linear interpolations.

Key properties: endpoints of the curve are P0 and Pn; the tangent directions at the endpoints align with the first and last control segments; the curve lies in the convex hull of its control points; the curve is invariant under affine transformations; adjustments to a control point influence the nearby region of the curve in a local way, with other points affected only through shared construction.

Common types include linear (n=1), quadratic (n=2), and cubic (n=3) Bézier curves; rational Bézier curves incorporate weights and can exactly represent conic sections. They are extended into NURBS to model complex shapes, allowing non-uniform parameterization and a combination of curve segments joined smoothly.

Applications include vector graphics, font design, digital typography, animation, and CAD/CAM systems, where smooth, scalable curves are required.