information to understand, test and make use of this model. Properties, theorems and construction methods are supplemented with algorithms and numerous examples and illustrations. Concerning the formalization, we have chosen to use precise and rigorous mathematical notations to remove any ambiguity and make understanding easier.
Readers unwilling to be concerned with mathematical formalisms can get to grips with the philosophy of our approach by focusing on the sections found at the beginnings of the chapters, in which ideas and principles are intuitively presented, based on examples.
This book is the result of 25 years of research carried out mainly in the LIRIS laboratories of the University of Lyon I and LIB of the University of Burgundy Franche-Comté. This research was initiated by Eric Tosan, who was instrumental at the origin of this formalism and to whom we dedicate this work.
Christian GENTIL
Gilles GOUATY
Dmitry SOKOLOV
February 2021
Introduction
I.1. Fractals for industry: what for?
This book shows our first steps toward the fundamental and applied aspects of geometric modeling. This area of research addresses the acquisition, analysis and optimization of the numerical representation of 3D objects.
Figure I.1. 3D tree built by iterative modeling
(source: project MODITERE no. ANR-09-COSI-014)
Figure I.1 shows an example of a structure that admits high vertical loads, while minimizing the transfer of heat between the top and bottom of the part. Additive manufacturing (3D printing) allows, for the first time, the creation of such complex objects, even in metal (here with a high-end laser printer EOS M270). This type of technology will have a high societal and economic impact, enabling better systems to be created (engines, cars, airplanes, etc.), designed and adapted numerically for optimal functionality, thus consuming less raw material, for their manufacturing, and energy, when used.
Current computer-aided design is, however, not well suited to the generation of such types of objects. For centuries, for millennia, humanity has produced goods with axes, files (or other sharp or planing tools), by removing bits from a piece of wood or plastic. Tools subsequently evolved into complex numerical milling machines. However, at no point during these manufacturing processes did we need sudden stops or permanent changes in the direction of the cutting tool. The patterns were always “regular”, hence the development of mathematics specific to these problems and our excellent knowledge of the modeling of smooth objects. This is why it was necessary to wait until the 20th century to have the mathematical knowledge needed to model rough surfaces or porous structures: we were just not able to produce them earlier.
Thus, since the development of computers in the 1950s, computer-aided geometric design (CAGD or CAD) software has been developed to represent geometric shapes intended to be manufactured by standard manufacturing processes. These processes are as follows:
– subtractive manufacturing, using machine tools such as lathes or milling machines;
– molding, where molds themselves are made using machine tools;
– deformation-based manufacturing: stamping or swaging (but again, dies are usually manufactured using machine tools), folding, etc.;
– cutting, etc.
Each of these processes imposes constraints, for example, concerning collision issues in milling machines (even a five-axis mill cannot produce any geometry). These manufacturing processes inevitably influenced the way we design the geometries of objects, in order to actually manufacture them. For example, CAD software has integrated these design methodologies by developing appropriate numerical models or tools. Currently, most CAD software programs are based on the representation of shapes by means of surfaces defining their edges. These surfaces are usually described using a parametric representation called non-uniform rational B-spline (NURBS). These surface models are very powerful and very practical. It is possible to represent any volume enclosed by a quadric (cylinders, cones, spheres, etc.) and complex shapes, such as car bodies or airplane wings. They were originally designed for this.
However, the emergence of additive manufacturing techniques has caused an upheaval in this area, opening up the possibility of potentially “manufacturable” forms. By removing the footprint constraint of the tool, it then becomes possible to produce very complex shapes with gaps or porosity. These new techniques have called into question the way objects are designed. New types of objects, such as porous objects or rough surfaces, can have many advantages, due to their specific physical properties. Fractal structures are used in numerous fields such as architecture (Rian and Sassone 2014), jewelry (Soo et al. 2006), heat and mass transport (Pence 2010), antennas (Puente et al. 1996; Cohen 1997) and acoustic absorption (Sapoval et al. 1997).
I.2. Fractals for industry: how?
The emergence of techniques such as 3D printers allows for new possibilities that are not yet used or are even unexplored. Different mathematical models and algorithms have been developed to generate fractals. We can categorize them into three families, as follows:
– the first groups algorithms for calculating the attraction basins of a given function. Julia and Mandelbrot (Peitgen and Richter 1986) or the Mandelbulb (Aron 2009) sets are just a few examples;
– the second is based on the simulation of phenomena such as percolation or diffusion (Falconer 1990);
– 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
