Bezier curve

Bézier curves are certain polynomials first described by the French engineer [Pierre Bézier]? who used them to design automobile bodies. The most important Bézier curves, the cubic ones, are used in several imaging systems such as PostScript and GIMP for drawing "smooth" curves.

Four points A, B, C and D in the plane or in three-dimensional space define a cubic Bézier curve. The curve starts at A going toward B and arrives at D coming from C. In general, it will not pass through B or C; these points are only there to provide directional information. The curve is always completely contained in the [convex hull]? of the four given points.

    B o         o C
         _____
     _,-'     `-._
   ,'             `.
  /                 \ 
 * A               D *

The parametric form of the curve is:

P(t) = A(1 - t)³ + 3Bt(1 - t)² + 3Ct²(1 - t) + Dt³    for 0 ≤ t ≤ 1.

The formula is inspired by the binomial distribution. Notice the binomial pattern in the coefficients [1, 3, 3, 1].

Generalizing the cubic case leads to higher order curves which require more than four control points; however, these do not find much use in practice.

See also: Splines?

References

• http://astronomy.swin.edu.au/pbourke/curves/bezier/