A Bézier curve is a curve expressed in the Bernstein basis:
Those 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:
- endpoints are interpolated
- The curves are tangent to end segments
- 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: