From splines to fractals

Deterministic splines and stochastic fractals are complementary techniques for generating free-form shapes. Splines are easily constrained and well suited to modeling smooth, man-made objects. Fractals, while difficult to constrain, are suitable for generating various irregular shapes found in nature. This paper develops constrained fractals, a hybrid of splines and fractals which intimately combines their complementary features. This novel shape synthesis technique stems from a formal connection between fractals and generalized energy-minimizing splines which may be derived through Fourier analysis. A physical interpretation of constrained fractal generation is to drive a spline subject to constraints with modulated white noise, letting the spline diffuse the noise into the desired fractal spectrum as it settles into equilibrium. We use constrained fractals to synthesize realistic terrain models from sparse elevation data.