Computing curves invariant under halving
Computer Aided Geometric Design - Special issue: Topics in CAGD
Fractals everywhere
Computer Aided Geometric Design
Towards free-form fractal modelling
Proceedings of the international conference on Mathematical methods for curves and surfaces II Lillehammer, 1997
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
The Fractal Nature of Bezier Curves
GMP '04 Proceedings of the Geometric Modeling and Processing 2004
Nonlinear subdivision through nonlinear averaging
Computer Aided Geometric Design
Aesthetic evolutionary algorithm for fractal-based user-centered jewelry design
Artificial Intelligence for Engineering Design, Analysis and Manufacturing
A Sufficient Condition for Uniform Convergence of Stationary p-Subdivision Scheme
Edutainment '08 Proceedings of the 3rd international conference on Technologies for E-Learning and Digital Entertainment
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Subdivision schemes generate self-similar curves and surfaces. Therefore there is a close connection between curves and surfaces generated by subdivision algorithms and self-similar fractals generated by Iterated Function Systems (IFS). We demonstrate that this connection between subdivision schemes and fractals is even deeper by showing that curves and surfaces generated by subdivision are also attractors, fixed points of IFS's. To illustrate this fractal nature of subdivision, we derive the associated IFS for many different subdivision curves and surfaces without extraordinary vertices, including B-splines, piecewise Bezier, interpolatory four-point subdivision, bicubic subdivision, three-direction quartic box-spline subdivision and Kobbelt's √3-subdivision surfaces. Conversely, we shall show how to build subdivision schemes to generate traditional fractals such as the Sierpinski gasket and the Koch curve, and we demonstrate as well how to control the shape of these fractals by adjusting their control points.