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Fractional half-tangent of a curve described by Iterated Function Systems.

Hicham Bensoudane 1 Christian Gentil 2, * Marc Neveu 2
* Corresponding author
Le2i - Laboratoire Electronique, Informatique et Image [UMR6306]
Abstract : The deterministic fractal curves and surfaces find many applications in modeling of rough objects. However, these curves and surfaces are nowhere differentiable. Without notion of tangent, we can not determine the relative orientation of two fractal shapes, to join them with a "natural" aspect. Various works proposed a generalization of the concept of derivative by introducing the fractional derivative. In this paper we apply this concept of fractional derivative to the curves described by Iterated Function Systems. We show that if the fractional derivative exists at boundary points of the curve, the direction of the fractional half-tangent is necessarily the eigenvector of the corresponding transformation of the IFS. From the property of self-similarity it is then possible to determine the fractional half tangents for a dense set of points. This concept is illustrated with some classical fractal curves.
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Submitted on : Saturday, January 11, 2014 - 3:42:56 PM
Last modification on : Monday, March 30, 2020 - 8:41:26 AM


  • HAL Id : hal-00927163, version 1


Hicham Bensoudane, Christian Gentil, Marc Neveu. Fractional half-tangent of a curve described by Iterated Function Systems.. Journal of Applied Functional Analysis, 2009, 4 (2), pp.311-326. ⟨hal-00927163⟩



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