Beziér

Beziér, often written Bézier, is a family of parametric curves used in computer graphics, typography, and geometric design. A Beziér curve is defined by a sequence of control points P0, P1, ..., Pn and a parameter t in the interval [0, 1]. The position on the curve is B(t) = sum_{i=0}^n binomial(n,i) (1−t)^{n−i} t^i P_i. The curve begins at P0 and ends at Pn; the intermediate control points influence the tangents and the overall shape. The curve lies within the convex hull of its control points.

Quadratic Beziér curves (n = 2) use three control points; cubic Beziér curves (n = 3) use four and are the most common in graphics and font design. In digital typography and vector graphics, cubic Beziér curves are standard in formats such as PostScript and SVG; TrueType fonts rely on quadratic Beziér curves.

Two key tools accompany Beziér curves: the de Casteljau algorithm, a numerically stable method for evaluating points on the curve and for splitting it into subcurves, and the concept of control polygons and the ability to join multiple curve segments to achieve complex shapes. Beziér curves are affine-invariant, and simple transformations preserve their shape.

Originating in the 1960s through the work of Pierre Bézier for Renault, the curves are also associated with Paul de Casteljau, who developed an algorithm for their computation. The spelling Beziér is a variant of the more common Bézier.