A Bézier curve is a curve expressed in the Bernstein basis:

Those points are called control points.

For n = 1, we just get a line segment. But for n = 3, we get “cubic Bézier”:

There are some nice features about the cubic Bézier curve:

  1. endpoints are interpolated
  2. The curves are tangent to end segments
  3. The curves are contained in convex hull (nice for rasterization)

The 3D version of a Bézier curve is a bézier patch.

Piecewise Bézier

Since high-degree Bernstein polynomials don’t interpolate well, we usually piece together many low-order Bézier curves. To get seamless curves, we need points and tangents to line up: