Bézier curve

A Bézier curve is a parametric curve used in computer graphics and related fields. It is named after French engineer Pierre Bézier, who used it in the 1960s for designing automobile bodies.

The mathematics behind Bézier curves use Bernstein polynomials to construct a smooth curve that passes through a set of control points. A cubic Bézier curve has four control points - two endpoints and two additional points that define the shape of the curve.

As the curve parameter moves from 0 to 1, the curve interpolates between the control points. The first and last control points are always the start and end points, but the middle points only guide the shape, they are not passed through. This allows the curve to follow a smooth, organic shape.

Bézier curves are popular because they are easy to use, editable through control points, and fast to evaluate. They are used for modeling smooth curves in 2D graphics, animations, typography, and 3D modeling. They are also used in path planning and trajectory generation applications. Different types of Bézier curves like quadratic or rational Béziers have evolved from the original cubic Béziers.

In summary, a Bézier curve provides a simple and flexible way to define curves using control points, with useful mathematical properties for graphics and design applications.