# Pseudo-Curvature of Fractal Curves for Geometric Control of Roughness

### Mohamad Janbein, Christian Gentil, Céline Roudet, Clement Poull

#### 2024

#### Abstract

Fractal geometry is a valuable formalism for synthesizing and analyzing irregular curves to simulate non-smooth geometry or roughness. Understanding and controlling these geometries remains challenging because of the complexity of their shapes. This study focuses on the curvature of fractal curves defined from an Iterated Function System (a set of contractive operators). We introduce the Differential Characteristic Function (DCF), a new tool for characterizing and analyzing their differential behavior. We associate a family of DCF to the fixed point of each operator. For each dyadic point of the curve, there exist left and right families of DCF inducing left and right ranges of curvatures: the pseudo-curvatures. A set of illustrations shows the influence of these pseudo-curvatures on the geometry of fractal curves. We propose a first approach for applying our results to roughness generation and control.

Download#### Paper Citation

#### in Harvard Style

Janbein M., Gentil C., Roudet C. and Poull C. (2024). **Pseudo-Curvature of Fractal Curves for Geometric Control of Roughness**. In *Proceedings of the 19th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP*; ISBN 978-989-758-679-8, SciTePress, pages 177-188. DOI: 10.5220/0012574800003660

#### in Bibtex Style

@conference{grapp24,

author={Mohamad Janbein and Christian Gentil and Céline Roudet and Clement Poull},

title={Pseudo-Curvature of Fractal Curves for Geometric Control of Roughness},

booktitle={Proceedings of the 19th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP},

year={2024},

pages={177-188},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0012574800003660},

isbn={978-989-758-679-8},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 19th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP

TI - Pseudo-Curvature of Fractal Curves for Geometric Control of Roughness

SN - 978-989-758-679-8

AU - Janbein M.

AU - Gentil C.

AU - Roudet C.

AU - Poull C.

PY - 2024

SP - 177

EP - 188

DO - 10.5220/0012574800003660

PB - SciTePress