Images of Julia sets that you can trust
João Batista Oliveira | de Figueiredo, Luiz Henrique | Nehab, Diego | Stolfi, Jorge
Julia sets | adaptive refinement | cell mapping | interval arithmetic
We present an algorithm for computing images of quadratic Julia sets that can be trusted in the sense that they contain numerical guarantees against sampling artifacts and rounding errors. We avoid point sampling by using interval arithmetic to classify entire rectangles in the complex plane. We avoid function iteration by using cell mapping and color propagation in graphs. This also avoids floating-point errors. As a result, our algorithm is able to robustly classify rectangles in the complex plane as being on either side of the Julia set. The union of the regions that cannot be so classified is guaranteed to contain the Julia set. Our algorithm computes a refinable quadtree decomposition of the complex plane adapted to the Julia set which can be used for rendering and for approximating geometric properties such as the area of the filled Julia set and the fractal dimension of the Julia set.