Geometric Modeling of Fractal Forms for CAD. Christian Gentil

      

Table of Contents

      1  Cover

      2  Title page

      3  Copyright

      4  Preface

      5  Introduction I.1. Fractals for industry: what for? I.2. Fractals for industry: how?

      6  1 The BC-IFS Model 1.1. Self-similarity and IFS 1.2. Controlled Iterated Function System 1.3. Boundary controlled iterated function system

      7  2 Design Examples 2.1. Curves 2.2. Wired structures 2.3. Surfaces and laces 2.4. Volumes and lacunar objects 2.5. Tree structures 2.6. Form assembly

      8  3 Surface NURBS, Subdivision Surfaces and BC-IFS 3.1. Bezier curves and surfaces 3.2. Uniform B-spline curves and surfaces 3.3. Generalization 3.4. NURBS curves 3.5. Subdivision curves and surfaces

      9  4 Building Operations, Assistance to Design and Applications 4.1. Topological consistency and symmetry constraints 4.2. Topological combination 4.3. Applications

      10  Conclusion

      11  Appendix: Data of Figures A.1. Data of figures A.2. Subdivision surface in Figure 3.6

      12  References

      13  Index

      14  Other titles from ISTE in Numerical Methods in Engineering

      15  End User License Agreement

      List of Illustrations

      1 IntroductionFigure I.1. 3D tree built by iterative modeling (source: project MODITERE no. AN...

      2 Chapter 1Figure 1.1. Schematic illustration of self-similarity. The black tree can be see...Figure 1.2. An example of a self-similar object composed of five copies of itsel...Figure 1.3. The self-similarity property, as shown in Figure 1.2, is symbolized ...Figure 1.4. Hausdorff distance. For a color version of this figure, see www.iste...Figure 1.5. Example of self-similarity involving non-contractive transformations...Figure 1.6. The Cantor set successively represented at the iteration levels from...Figure 1.7. Cartesian product of two Cantor sets successively represented at ite...Figure 1.8. The Sierpinski triangle successively represented at iteration levels...Figure 1.9. The Menger sponge successively represented at iteration levels from ...Figure 1.10. Example of Romanesco broccoli consisting of seven self-similar elem...Figure 1.11. On the left-hand side, we provide a few examples of self-similarity...Figure 1.12. Example of a decomposition of an L-shape into several similar eleme...Figure 1.13. Example of self-similarity. The object on the left-hand side has a ...Figure 1.14. Lattice structure of the attractors. On the left, the lattice struc...Figure 1.15. An example of a connection between two attractors. The green attrac...Figure 1.16. The evaluation tree of the attractor of the IFS computed at the thi...Figure 1.17. Example of the parameterization of the attractor in Figure 1.13. On...Figure 1.18. Example of a transport mapping that defines a morphism of IFS. For ...Figure 1.19. Example of mapping between two attractors using the transport map. ...Figure 1.20. Attractor defining the parameter space for the Sierpinski triangle ...Figure 1.21. Automaton of an IFS

. The transition i is associated with the tran...Figure 1.22. Example of a three-state automaton inducing a restriction of the se...Figure 1.23. Both images represent the attractors defined from the same transfor...Figure 1.24. Other examples of attractors built from the same automatons as thos...Figure 1.25. Automaton generating the union of two attractors. Transitions 0 and...Figure 1.26. The internal structure of the Menger sponge. On the left, the inter...Figure 1.27. Automaton describing the structure of the image on the right-hand s...Figure 1.28. Example of a two-state automaton: the □ is divided into four △ and ...Figure 1.29. Construction of the sequence converging to the attractor of the aut...Figure 1.30. Approximation