Family of recursively defined curves, related to the cubic Bézier curve

Starting from elementary geometrical, combinatorial and algorithmic properties of recursively defined curves, a parametrized class of curves is defined that incorporates the cubic Bézier curve together with a continuous set of generalizations, and also a parametrized set of fractal curves. The shape of the latter may be controlled in a similar fashion to the smooth Bézier-like curves. Both the smooth and the fractal curves lead to a straightforward extension towards surface definitions.