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:
Bézier curves are attractive in computer graphics for two main reasons:
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. Instead, complicated curves are pieced together from cubic curves: the first has control points A, B, C, and D, the second has control points D, E, F, and G, and to ensure smoothness at D, one requires D-C = E-D.
See also: Splines?, [Bernstein polynomial]?, [Bézier surface]?