Section: New Results

Fractal Geometry

Participant : Dmitry Sokolov.

Fractal geometry is a relatively new branch of mathematics that studies complex objects of non-integer dimensions. It finds applications in many branches of science as objects of such complex structure often exhibit interesting properties.

In 1988 Barnsley presented the Iterative Function System (IFS) model that allows modelling complex fractal shapes with only a limited set of contractive transformations. Later many other models were based on the IFS model such as Language-Restricted IFS, Projective IFS, Controlled IFS and Boundary Controlled IFS. The latter allows modeling complex shapes with control points and specific topology. These models cover classical geometric models such as B-splines and subdivision surfaces as well as fractal shapes.

This year we focused on the analysis of the differential behaviour of the shapes described with Controlled IFS and Boundary Controlled IFS. We derive the necessary and sufficient conditions for differentiability for everywhere dense sets of points. Our study is based on the study of the eigenvalues and eigenvectors of the transformations composing the IFS.

We apply the obtained conditions to modeling curves in surfaces. We describe different examples of differential behaviour presented in shapes modeled with Controlled IFS and Boundary Controlled IFS. We also use the Boundary Controlled IFS to solve the problem of connecting different subdivision schemes. We construct a junction between Doo-Sabin and Catmull-Clark subdivision surfaces and analyse the differential behaviour of the intermediate surface.

An article about this work is in the publication process in LNCS.