Geometric Modeling of Fractal Forms for CAD. Christian Gentil

       – the last corresponds to deterministic or probabilistic algorithms or models based on the self-similarity property associated with fractals such as the terrain generator (Zhou et al. 2007), the iterated function system (Barnsley et al. 2008) or the L-system (Prusinkiewicz and Lindenmayer 1990).

      In the latter family of methods, shapes are generated from rewriting rules, making it possible to control the geometry. Nevertheless, most of these models have been developed for image synthesis, with no concerns for “manufacturability”, or have been developed for very specific applications, such as wood modeling (Terraz et al. 2009). Some studies approach this aspect for applications specific to 3D printers (Soo et al. 2006). In (Barnsley and Vince 2013b), Barnsley defines fractal homeomorphisms of [0, 1]² onto the modeling space [0, 1]². The same approach is used in 3D to build 3D fractals. A standard 3D object is integrated into [0, 1]³ and then transformed into a 3D fractal object. This approach preserves the topology of the original object, which is an important point for “manufacturability”.

      The main difficulty associated with traditional methods for generating fractals lies in controlling the forms. For example, it is difficult to obtain the desired shape using the fractal homeomorphism system proposed by Barnsley. Here, we develop a modeling system of a new type based on the principles of existing CAD software, while expanding their capabilities and areas of application. This new modeling system offers designers (engineers in industry) and creators (visual artists, designers, architects, etc.) new opportunities to quickly design and produce a model, prototype or unique object. Our approach consists of expanding the possibilities of a standard CAD system by including fractal shapes, while preserving ease of use for end users.

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.