Section: New Results

Control of the differential behaviour of the joining curve between two fractal curves

Participants : Dmitry Sokolov, S. Podkorytov, C. Gentil, S. Lanquetin.

The general objective of our work is to create a geometric modeller based on iterative processes. Iterative processes can be used to describe a wide array of shapes inaccessible to standard methods such as fractal curves or sets. Our work is based on Boundary Controlled Iterative System (BCIFS). BCIFS upgrades the standard iterative process such as Iterated Function System (IFS) with B-Rep structure. We can describe objects with familiar B-rep structure, where each cell is a fractal object. For instance, if we consider a polyhedron, then each face is a fractal surface, and each edge is a fractal curve. Objects modelled with BCIFS not necessary have the fractal properties, objects such as B-splines curves and surfaces can be modelled as well. So with BCIFS formalism we can operate with both standard and fractal objects.

With this objective in mind, we have to provide tools that work with fractal objects in the same manner as with objects of classical topology. In this project we focus on the constructing of an intermediate curve between two other curves defined by different iterative construction processes. Similar problem often arises with subdivision surfaces, when the goal is to connect two surfaces with different subdivision masks. We start by dealing with curves, willing to later generalize our approach to surfaces. We formalise the problem with Boundary Controlled Iterated Function System model. Then we deduct the conditions that guaranties continuity of the intermediate curve. These conditions determine the structure of subdivision matrices. By studying the eigenvalues of the subdivision operators, we characterise the differential behaviour at the connection points between the curves and the intermediate one. This behaviour depends on the nature of the initial curves and coefficients of the subdivision matrices. We also suggest a method to control the differential behaviour by adding intermediate control points (Figure 7 ). This work was presented at the Symposium on Solid and Physical Modeling [23] .

Figure 7. Two intermediate curves between the fractal curve and B-spline. Three control point are used to control the shape of the curve