Bézier curve

Did You Know?

• The mathematical basis for Bézier curves—the Bernstein polynomials—was established in 1912, but not applied to graphics until 1959 when Paul de Casteljau developed de Casteljau's algorithm at Citroën.
• De Casteljau's method was patented in France but not published until the 1980s, while Pierre Bézier independently discovered and widely publicised the curves in the 1960s at Renault.
• A Bézier curve of degree n can be converted to degree n+1 with the same shape, useful for software that only supports cubic curves.

Specific cases

A Bézier curve is defined by control points P0 through Pn, where n is the order (1 for linear, 2 for quadratic, 3 for cubic, etc.). The first and last points are the endpoints; intermediate points generally do not lie on the curve. A linear Bézier curve is simply a line between P0 and P1, equivalent to linear interpolation. A quadratic Bézier curve, with points P0, P1, and P2, can be seen as the linear interpolant of corresponding points on two linear Bézier curves; its tangents at P0 and P2 intersect at P1. A cubic Bézier curve uses four points P0, P1, P2, P3; it starts at P0 toward P1 and ends at P3 from the direction of P2, with P1 and P2 providing directional information. The distance between P1 and P2 influences how the curve moves toward P1 before turning. The cubic curve can be expressed as an affine combination of two quadratic Bézier curves. For some choices of P1 and P2, the curve may intersect itself or contain a cusp. Any series of four distinct points can be converted to a cubic Bézier curve that passes through all four in order.

General definition

Bézier curves exist for any degree n. A degree-n curve can be defined recursively as a linear interpolation between corresponding points on two degree-(n−1) curves. The explicit formula involves binomial coefficients and Bernstein basis polynomials of degree n. The points Pi are called control points, and connecting them in order creates the Bézier polygon, also known as the control polygon. The curve lies within the convex hull of this polygon. The curve starts at P0 and ends at Pn, a property called endpoint interpolation. It is a straight line only when all control points are collinear. The curve’s beginning and end are tangent to the first and last segments of the Bézier polygon, respectively. Any Bézier curve can be split at a point into two subcurves, each also a Bézier curve. The curve can be rewritten as a polynomial, but high-degree versions may suffer from numeric instability; in such cases, de Casteljau’s algorithm is recommended.

Applications

Bézier curves are widely used in computer graphics to model smooth curves. Because the curve lies within the convex hull of its control points, the points can be displayed and used to manipulate the curve intuitively. Affine transformations such as translation and rotation can be applied by transforming the control points. Quadratic and cubic Bézier curves are most common; higher-degree curves are more computationally expensive. For complex shapes, low-order Bézier curves are patched together into a composite Bézier curve, commonly called a 'path' in vector graphics languages, standards, and programs. To join curves without kinks, G1 continuity forces the meeting control point to lie on the line defined by the two adjacent control points. The simplest rasterization method evaluates the curve at many points and scan converts line segments; an adaptive method is recursive subdivision, checking if the curve approximates a line within tolerance and subdividing if not. Forward differencing methods exist but require careful error analysis. Analytical intersection with scan lines is rarely used due to complexity.