Geometric Modeling of Fractal Forms for CAD. Christian Gentil

of Figure 1.28 obtained wi...Figure 1.31. Evaluation tree developed at level 2, for the attractor of the auto...Figure 1.32. Example of a third-degree B-spline surface defined from a grid of c...Figure 1.33. The surface, at the top right, is a smooth B-spline surface and has...Figure 1.34. Example of a curve constructed based on an FIF. The parallelepipeds...Figure 1.35. Barycentric space. On the left, the barycentric space of dimension ...Figure 1.36. Cantor set built in the barycentric space BI² using the IFS compose...Figure 1.37. Sierpinski triangle built in the barycentric space BI³. For a color...Figure 1.38. Example of projections of the Sierpinski triangle. The attractor is...Figure 1.39. Example of a two-state automaton. The □ is divided into three □ and...Figure 1.40. Three different projections of the attractor described by the autom...Figure 1.41. Automaton defining the attractor in the barycentric spaces and perf...Figure 1.42. Curve of the “Takagi” type, defined from three control points and t...Figure 1.43. Incidence constraints. On the left: Three curves of the “Takagi” ty...Figure 1.44. Example of the construction of a connection between the subdivision...Figure 1.45. Automaton integrating the cellular decomposition of a curve subdivi...Figure 1.46. Tree for a curve. For a color version of this figure, see www.iste....Figure 1.47. Quotient graph for a curve. For a color version of this figure, see...Figure 1.48. Example of curves generated for different parameter values. For a c...Figure 1.49. Attractor built from the IFS

whose subdivision operators correspo...Figure 1.50. Subdivision structure of the tile. For a color version of this figu...Figure 1.51. Automaton for the subdivision of a quadrangular surface. For a colo...Figure 1.52. Cell structure of a quadrangular tile. For a color version of this ...Figure 1.53. Example of a quadrangular surface. For a color version of this figu...Figure 1.54. Example of a quadrangular surface bordered by Bezier curves with an...Figure 1.55. Example of a quadrangular surface structure bordered by Bezier curv...Figure 1.56. Example of a surface structure with fractal topology, obtained from...Figure 1.57. Example of the quadrangular surface bordered by Bezier curves with ...Figure 1.58. Example of curves projected into the modeling space

, following th...Figure 1.59. Example of curves projected into the modeling space

, following th...Figure 1.60. Example of curves projected into the modeling space

, following th...Figure 1.61. Example of a network of control points for a triangular surface tha...

      3 Chapter 2Figure 2.1. “Standard” automaton for a curve. Transitions referred to by the sym...Figure 2.2. Examples of curves constructed using two transformations. The contro...Figure 2.3. A tree illustrating another possibility for connecting the subdivisi...Figure 2.4. Comparison of the effect of the two types of connection, standard an...Figure 2.5. Examples of curves constructed using three transformations whose ver...Figure 2.6. Examples of von Koch curves constructed with two transformations (to...Figure 2.7. Left-hand column: for the first figure, the first subdivision point ...Figure 2.8. Automaton describing a CA curve subdivided into a CA-type curve and ...Figure 2.9. Example of a curve obtained with two mutually referenced states acco...Figure 2.10. Tree illustrating the construction of a connection for a wired stru...Figure 2.11. Example of two wired structures built from the incidence and adjace...Figure 2.12. Examples of wired structures. The diagrams above each shape represe...Figure 2.13. Diagram of the cellular decomposition of a quadrangular surface wit...Figure 2.14. Example of subdivision of a quadrangular surface with “non-standard...Figure 2.15. Connections to build a Hilbert/Peano curve. The red circles show th...Figure 2.16. Example of a curve attempt that satisfies the adjacency relations i...Figure 2.17. Subdivision of a quadrangular surface satisfying the constraints of...Figure 2.18. Example of a Hilbert/Peano surface, defined from the subdivision of...Figure 2.19. The diagram on the right-hand side presents the quadrangular subdiv...Figure 2.20. To achieve a quadrangular surface from two subdivisions, we need to...Figure 2.21. Automaton symbolizing the subdivision system of a quadrangular surf...Figure 2.22. Example of a quadrangular surface with two subdivisions. For this i...Figure 2.23. Standard triangular subdivision. The triangular face is subdivided ...Figure 2.24. Standard triangular subdivision, but with connections differing fro...Figure 2.25. Pentagonal subdivision. On the left, the incidence relations are sy...Figure 2.26. On the left, incidence relations are symbolized using red dotted li...Figure 2.27. Cellular decomposition and subdivision of the Sierpinski triangle. ...Figure 2.28. Cellular decomposition and subdivision of the Sierpinski triangle w...Figure 2.29. Example of a Sierpinski triangle whose face, edges and vertices hav...Figure 2.30. Example of a Sierpinski triangle whose edges are uniform quadratic ...Figure 2.31. Penrose tiling of the “kite” type at iterations 1, 2, 3 and 6. For ...Figure 2.32. Topological subdivision diagram of the faces and edges representing...Figure 2.33. Example of 3D surfaces constructed from the topological subdivision...Figure 2.34. Examples of the Menger sponge, represented with three iteration lev...Figure 2.35. Example of the construction of a 2D tree structure, consisting of t...Figure 2.36. Subdivision process of the tree structure of Figure 2.35 and cell d...Figure 2.37. Automaton describing the iterative construction process of the tree...Figure 2.38. Examples of projection of the tree’s topological structure, defined...Figure 2.39. Subdivision process of a tree structure whose trunk is subdivided i...Figure 2.40. Automaton describing the iterative construction process of the tree...Figure 2.41. Example of a tree structure whose trunk (green) is not subdivided (...Figure 2.42. Simplified representation of the incidence and adjacency relations ...Figure 2.43. 3D tree built on the principle of space tiling(source: project MODI...Figure 2.44. Example of assembling fractal structures (built from an octagonal f...Figure 2.45. Example of an assembly of triangular surface structures. The basic ...Figure 2.46. Examples of assemblies of pentagonal fractal faces following a dode...Figure 2.47. Assembly of 10 3D Penrose tilings of the “kite”-type to form the co...Figure 2.48. Examples of assemblies built from Menger sponges and manufactured b...

      4 Chapter 3Figure 3.1. On the left is an example of a quadratic Bezier curve, defined by th...Figure 3.2. C-IFS automaton whose attractor is a Bezier curve. This automaton si...Figure 3.3. Illustration of the self-similarity property of uniform quadratic B-...Figure 3.4. Blending B-spline functions of the second degree. Their support is o...Figure 3.5. The control points (P₀, P₁, P₂, P₃) and the knot vector (u₀, u₁, u₂,...Figure 3.6. Subdivision scheme obtained by duplication of knotsFigure 3.7. Automaton of the C-IFS representing the iterative process of a secon...Figure 3.8. From left to right: The non-uniform quadratic curve defined from fou...Figure 3.9. Automaton of the C-IFS representing the subdivision of a third-degre...Figure 3.10. The top diagram represents the knot interval vector of a curve of d...Figure 3.11. Illustration of the extraction of the knot vectors from the two cur...Figure 3.12. Subdivision process of a NURBS surface of degree 2, obtained by the...Figure 3.13. Automaton of the C-IFS representing the subdivision of a NURBS surf...Figure 3.14. C-IFS automaton whose attractor is a subdivision curve built from t...Figure 3.15. Approximations of a uniform cubic B-spline curve for levels 0 to 6:...Figure 3.16. Refinement of a regular control mesh. In red: Minimal control mesh ...Figure 3.17. Self-similarity of the refined mesh. Each of the four blue sub-mesh...Figure 3.18. Automaton of a subdivision surface for the Doo–Sabin schemeFigure 3.19. Refinement of an irregular control mesh. In red: Minimal control me...Figure 3.20. Self-similarity of the refined mesh. In blue, the four sub-meshes o...Figure 3.21. Automaton of a surface subdivision for the Doo–Sabin scheme with an...Figure 3.22. Adjacency and incidence constraints. On the left: Example of an adj...Figure 3.23. Control points defining one of the irregular edges. The irregular e...Figure 3.24. Illustration of incidence and adjacency constraints for an irregula...Figure 3.25. Representation of the topological subdivision process of an irregul...Figure 3.26. Cell decomposition of an irregular tile. The constraints are repres...Figure 3.27. Subdivision of an irregular tile with an irregular face of six side...Figure 3.28. Catmull subdivision of a regular tileFigure 3.29. The four sub-meshes of 4 × 4 control points are represented in blue...Figure 3.30. Example of an irregular mesh with a vertex of valence five. After r...Figure 3.31. Illustration of the Loop subdivision. Figure 3.31(a) is an illustra...Figure 3.32. Refinement of a regular control mesh for the Loop scheme. The struc...Figure 3.33. Self-similarity of the refined mesh for the Loop scheme. Decomposit...

      5 Chapter