Subdivision Schemes and Attractors

Abstract

Subdivision schemes generate self-similar curves and surfaces. Therefore there is a close connection between curves and surfaces generated by subdivision algorithms and self-similar fractals generated by Iterated Function Systems (IFS). We demonstrate that this connection between subdivision schemes and fractals is even deeper by showing that curves and surfaces generated by subdivision are also attractors, fixed points of IFS's. To illustrate this fractal nature of subdivision, we derive the associated IFS for many different subdivision curves and surfaces without extraordinary vertices, including B-splines, piecewise Bezier, interpolatory four-point subdivision, bicubic subdivision, three-direction quartic box-spline subdivision and Kobbelt's p3-subdivision surfaces. Conversely, we shall show how to build subdivision schemes to generate traditional fractals such as the Sierpinski gasket and the Koch curve, and we demonstrate as well how to control the shape of these fractals by adjusting their control points.