Bezier

Bezier curves are parametric curves widely used in computer graphics and geometric design. They are defined by an ordered set of control points P0 through Pn. The curve is expressed as B(t) = sum_{i=0}^n P_i B_i^n(t) for t in [0,1], where B_i^n(t) = binomial(n,i) (1-t)^{n-i} t^i are Bernstein polynomials. The endpoints P0 and Pn lie on the curve, and the interior points influence its shape through the tangents at the ends and the overall curvature.

Key properties include affine invariance—applying a linear transformation to the control points yields the corresponding transformation of the curve—and the convex hull property, which states the curve lies within the convex hull of its control points. The derivative at the ends is proportional to P1−P0 and Pn−P_{n−1}, determining the tangent directions. The parameter t provides a smooth progression along the curve, and the curve can be subdivided or evaluated efficiently.

Evaluation and subdivision are performed by de Casteljau’s algorithm, which recursively linearly interpolates between control points. The algorithm is numerically stable and can produce exact subdivisions into two Bezier curves that together reproduce the original curve.

Bezier curves come in various degrees; linear (degree 1), quadratic (degree 2), and cubic (degree 3) are most common. Higher-degree forms exist but are less practical for real-time rendering. They are widely used in font design, vector graphics formats such as SVG (cubic Bezier paths), CAD, and animation, where control over smooth shapes and scalable curves is essential.

Historically, the curves are named after Pierre Bézier, who popularized them in the 1960s for automobile design, while the underlying algorithm was developed by Paul de Casteljau at Citroën in 1959.