Geometric Modeling of Fractal Forms for CAD

Designing and controlling complex shapes like porous volumes and rough surfaces is a challenge. Fractal geometry is an interesting approach which considerably simplify the problem. Even though underlying concepts reduce the set possible shapes, they generate a surprising variety of shapes.

In this book we present a formalism to design such complex objects for geometric aided geometry design applications. The goal of this formalism is to provide to the end user the possibility to manipulate fractal objects as a standard euclidean object with standard tools of CAD system. This formalism encompass curves, surfaces, volumes, as well as NURBS and subdivision surfaces. All theoretical and practical aspects are developed, from the design up to 3D printing.

1 Introduction to fractal geometry

2 Model BCIF

2.1 Iterated Function System (IFS)

2.2 Controlled Iterated Function System (CIFS)

2.3 Boundary Controlled Iterated Function System (BCIFS)

3 Example of design

3.1 curves and wire-frames

3.2 surfaces and laces

3.3 volumes and porous objects

3.4 Trees

4 Topological combination

5 Efficient approximation for 3D printing

6 NURBS and Subdivision surfaces and (BCIFS)

7 Applications