Béziercurve

A Bézier curve is a parametric curve widely used in vector graphics and computer graphics. A Bézier curve of degree n is defined by n+1 control points P0, P1, ..., Pn in a Euclidean space. The curve is traced by the function B(t) for t in [0, 1], with B(0) = P0 and B(1) = Pn.

The position on the curve at parameter t is given by B(t) = sum_{i=0}^n B_{i,n}(t) P_i, where B_{i,n}(t) are Bernstein polynomials B_{i,n}(t) = C(n,i) t^i (1−t)^{n−i}. Equivalently, B(t) can be computed by the De Casteljau algorithm, which forms successive linear interpolations between control points to yield a point on the curve for any t.

Key properties include affine invariance (linear transformations of the control points translate to the same transformation of the curve) and the convex hull property (the curve lies within the convex hull of its control points). The endpoints and tangents are determined by the outer control points: the tangent at t = 0 is proportional to P1 − P0, and the tangent at t = 1 is proportional to Pn − P_{n−1}. Bézier curves support subdivision, meaning a single curve can be divided into two Bézier curves with new control points derived from De Casteljau’s construction.

Extensions and variants include rational Bézier curves, which incorporate positive weights to represent a wider range of shapes, including conic sections. Cubic Béziers are particularly common in fonts and vector graphics. Applications span font design, scalable vector graphics, animation, and interactive modeling.

Historically, Bézier curves are named after Pierre Bézier, who popularized their use in automotive design, with foundational algorithms developed by Paul de Casteljau in the 1950s.