%0 Journal Article
%T Fractional half-tangent of a curve described by Iterated Function Systems.
%+ Laboratoire Electronique, Informatique et Image [UMR6306] (Le2i)
%+ Modélisation géométrique et synthèse d’images (MGSI)
%A Bensoudane, Hicham
%A Gentil, Christian
%A Neveu, Marc
%< avec comité de lecture
%J Journal of Applied Functional Analysis
%V 4
%N 2
%P 311-326
%8 2009-04-01
%D 2009
%K fractional dérivative
%K fractal
%K self-similarity
%K tangent
%K iterated function sustems
%Z Computer Science [cs]/Computational Geometry [cs.CG]
%Z Computer Science [cs]/Modeling and Simulation
%Z Computer Science [cs]/Graphics [cs.GR]
%Z Mathematics [math]/Differential Geometry [math.DG]Journal articles
%X 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.
%G English
%L hal-00927163
%U https://u-bourgogne.hal.science/hal-00927163
%~ UNIV-BOURGOGNE
%~ CNRS
%~ ENSAM
%~ INSMI
%~ LE2I
%~ TDS-MACS
%~ AGREENIUM
%~ LIB_UB
%~ LIB_MODGEOM
%~ ARTS-ET-METIERS-SCIENCES-ET-TECHNOLOGIES
%~ INSTITUT-AGRO