Preprint A721/2012
Images of Julia sets that you can trust

João Batista Oliveira | de Figueiredo, Luiz Henrique | Nehab, Diego | Stolfi, Jorge

**Keywords: **
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.