Bezier Curves and Surfaces

Bezier curves emerged in the 1960s in the context of industrial design (Renault, Pierre Bezier). They rely on a simple and powerful geometric idea: a curve or surface is obtained as a convex combination of control points through Bernstein polynomials.

In its most common form, a curve of degree $n$ is given by

\[B(t) = \sum_{i=0}^{n} \binom{n}{i}(1-t)^{n-i}t^i\,P_i,\qquad t\in[0,1].\]

and can be evaluated in a numerically stable way with De Casteljau’s algorithm, which repeatedly performs linear interpolations between consecutive points.

For surfaces, a tensor product with two parameters $(u,v)\in[0,1]^2$ is used:

\[S(u,v)=\sum_{i=0}^{n}\sum_{j=0}^{m} B_n^i(u)\,B_m^j(v)\,P_{i,j}.\]

How to Use the Interactive

• 2D curve: click to add a point, Shift+click to delete, drag to move.
• 3D curve and 3D surface: to move points, first turn off Camera mode (when it is active, the mouse controls orbit and zoom).