The Marriage of Fractals and Splines

Google Tech Talk June 15, 2009 ABSTRACT The Marriage of Fractals and Splines: Fractals with Control Points, Splines as Attractors Presented by Ron Goldman, Department of Computer Science, Rice University. Fractals and splines have very different geometric features. Fractals can be continuous everywhere, yet differentiable nowhere. Fractals are often selfsimilar curves with fractional dimension. And fractals are also attractors, fixed points of iterated function systems. In contrast, splines are piecewise polynomial curves, so well behaved that they are often used for large scale industrial design and manufacture. Splines are essentially polynomials, so splines are onedimensional curves that are differentiable everywhere. Splines have control points -- polynomial coefficients -- that can be used to control the shape of the spline in an intuitive fashion. Moreover, unlike fractals, splines have parametrizations. Nevertheless, the goal of this talk is to marry fractals and splines: to demonstrate that fractals and splines share many geometric properties and algorithms. We shall show that just like splines, fractals can be parametrized and fractals have control points that allow us to adjust the shape of the fractal in an intuitive manner. Moreover, just like fractals, splines are attractors, fixed points of iterated function systems. We shall show how to apply fractal algorithms to generate splines and spline algorithms to generate fractals. We conclude that fractals and splines are not really that different after all.