Christian Gentil - Geometric Modeling of Fractal Forms for CAD

We propose a formalism based on standard iterated function systems (IFS) enhanced by the concept of boundary representation (B-rep). This makes it possible to separate the topology of the final forms from the geometric texture, which greatly simplifies the design process. This approach is powerful, and it generalizes subdivision curves and standard surfaces (linear, stationary), allowing for additional control. For example, we have been able to propose a method for connecting a primal subdivision scheme surface with a dual subdivision scheme surface (Podkorytov et al . 2014), which is a difficult subject for the standard subdivision approach.

The first chapter recalls the notion of self-similarity, intimately linked to that of fractality. We present the IFS, formalizing this property of self-similarity. We then introduce enhancements into this model: controlled iterated function systems (C-IFS) and boundary controlled iterated function systems (BC-IFS). The second chapter is devoted to examples. It provides an overview of the possibilities of description and modeling of BC-IFS, but also allows better understanding the principle of the model through examples. The third chapter presents the link between BC-IFS, the NURBS surface model and subdivision surfaces. The results presented in this chapter are important because they show that these surface models are specific cases of BC-IFS. This allows them to be manipulated with the same formalism and to make them interact by building, for example, junctions between two surfaces of any kind. In the fourth chapter, we outline design tools that facilitate the description process, as well as examples of the applications, of the design of porous volumes and rough surfaces.

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