Papers by Christian Gentil
Second Derivative and Curvature of Fractal Curves
Pseudo-Curvature of Fractal Curves for Geometric Control of Roughness
Chaos Solitons & Fractals, Nov 1, 2012
HAL (Le Centre pour la Communication Scientifique Directe), Nov 23, 2022
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
HAL (Le Centre pour la Communication Scientifique Directe), Oct 29, 2012
The general objective of our work is to create a geometric modeller based on iterative processes.... more The general objective of our work is to create a geometric modeller based on iterative processes. With this objective in mind, we have to provide tools that work with fractal objects in the same manner as with objects of classical topology. In this article we focus on the constructing of an intermediate curve between two other curves defined by different iterative construction processes. Similar problem often arises with subdivision surfaces, when the goal is to connect two surfaces with different subdivision masks. We start by dealing with curves, willing to later generalize our approach to surfaces. We formalize the problem with Boundary Controlled Iterated Function System model. Then we deduct the conditions that guaranties continuity of the intermediate curve. These conditions determine the structure of subdivision matrices. By studying the eigenvalues of the subdivision operators, we characterize the differential behaviour at the connection points between the curves and the intermediate one. This behaviour depends on the nature of the initial curves and coefficients of the subdivision matrices. We also suggest a method to control the differential behaviour by adding intermediate control points.
Méthodes de contrôle de la rugosité à partir des propriétés différentiels de courbes autosimilaires
HAL (Le Centre pour la Communication Scientifique Directe), Mar 15, 2023
Représentation, analyse et caractérisation de surfaces rugueuses
HAL (Le Centre pour la Communication Scientifique Directe), Jul 2, 2020
NURBS and Iterated Functions Systems
International audienc
Nous nous intéressons au calcul des tangentes à une courbe fractale définie à l'aide d'un... more Nous nous intéressons au calcul des tangentes à une courbe fractale définie à l'aide d'un IFS. Généralement, les courbes fractales sont nulle part dérivables, mais sous certaines conditions on peut montrer qu'elles admettent, en un ensemble de points, des demi-tangentes à droite et à gauche. Nous proposons une méthode permettant de déterminer ces demi-tangentes.
In this paper, we present an algorithm to construct an approximate convex hull of the attractors ... more In this paper, we present an algorithm to construct an approximate convex hull of the attractors of an affine iterated function system (IFS). We construct a sequence of convex hull approximations for any required precision using the self-similarity property of the attractor in order to optimize calculations. Due to the affine properties of IFS transformations, the number of points considered in the construction is reduced. The time complexity of our algorithm is a linear function of the number of iterations and the number of points in the output convex hull. The number of iterations and the execution time increases logarithmically with increasing accuracy. In addition, we introduce a method to simplify the approximation of the convex hull without loss of accuracy.
Abstract. The aim of our work is to specify and develop a geometric modeler, based on the formali... more Abstract. The aim of our work is to specify and develop a geometric modeler, based on the formalism of iterated function systems with the following objectives: access to a new universe of original, various, aesthetic shapes, modeling of conventional shapes (smooth surfaces, solids) and unconventional shapes (rough surfaces, porous solids) by defining and controlling the relief (surface state) and lacunarity (size and distribution of holes). In this context we intend to develop differential calculus tools for fractal curves and surfaces defined by IFS. Using local fractional derivatives, we show that, even if most fractal curves are nowhere differentiable, they admit a left and right half-tangents, what gives us an additional parameter to characterize shapes.
Propriétés différentielles du raccord entre deux courbes fractales
International audienc
Description of fractals with IFS
Le but de notre travail est d'élaborer une méthode de construction des formes paramétrées (courbe... more Le but de notre travail est d'élaborer une méthode de construction des formes paramétrées (courbes, surfaces, ...) dont l'aspect local est variable et non uniforme : à chaque point sera associé une « texture géométrique » pouvant passer continument du lisse au rugueux.
Mixed-aspect fractal surfaces
Computer-Aided Design, 2013
ABSTRACT In order to provide accurate tools to model original surfaces in a Computer Aided Geomet... more ABSTRACT In order to provide accurate tools to model original surfaces in a Computer Aided Geometric Design context, we develop a formalism based on iterated function systems. This model enables us to represent both smooth and fractal free-form curves and surfaces. But, because of the self-similarity property underlying the iterated function systems, curves and surfaces can only have homogeneous roughness. The aim of our work was to elaborate a method to build parametric shapes (curves, surfaces, …) with a non-uniform local aspect: every point is assigned a “geometric texture” that evolves continuously from a smooth to a rough aspect. The principle is to blend shapes with uniform aspects to define a shape with a variable aspect. A blending function controls the influence of each initial shape. An illustrated application is then built, joining surfaces characterized by different kinds of roughness.
Chaos, Solitons & Fractals, 2012
In this paper, we present an algorithm to construct an approximate convex hull of the attractors ... more In this paper, we present an algorithm to construct an approximate convex hull of the attractors of an affine iterated function system (IFS). We construct a sequence of convex hull approximations for any required precision using the self-similarity property of the attractor in order to optimize calculations. Due to the affine properties of IFS transformations, the number of points considered in the construction is reduced. The time complexity of our algorithm is a linear function of the number of iterations and the number of points in the output convex hull. The number of iterations and the execution time increases logarithmically with increasing accuracy. In addition, we introduce a method to simplify the approximation of the convex hull without loss of accuracy.
HAL (Le Centre pour la Communication Scientifique Directe), Dec 1, 2015
Attempts to model gases in computer graphics started in the late 1970s. Since that time, there ha... more Attempts to model gases in computer graphics started in the late 1970s. Since that time, there have been many approaches developed. In this paper we present a non-physical method allowing to create vapourish objects like clouds or smoky characters. The idea is to create few sketches describing the rough shape of the final vapourish object. These sketches will be used as condensation sets of Iterated Function Systems, providing intuitive control over the object. The advantages of the new method are: simplicity, good control of resulting shapes and ease of eventual object animation.
WSCG 2018 - Full papers proceedings
We present a new representation of uniform subdivision surfaces based on Iterated Functions Syste... more We present a new representation of uniform subdivision surfaces based on Iterated Functions Systems formalism. Main advantages of this new representation are the formalization of topological subdivision, multiscale representation of limit surface, separation of iterative space where the attractor is computed once for all and modeling space where the attractor is projected many times. An important consequence of this approach is that all uniform subdivision schemes are handled in the same way whatever there are primal or dual, approximating or interpolating. Subdivision surfaces are no longer viewed as a set of rules but as a list of barycentric combinations to apply on neighborhoods of the coarse mesh. These combinations are representative subsets of the attractor which is deduced from a Controlled Iterated Functions System automaton. From this new point of view we present in this paper a straightforward implementation to directly compute a tessellation of the subdivision surface from a control mesh. This implementation takes full advantage of Graphics Processing Units high capability of computation and Tessellation Stage of OpenGL/GLSL rendering pipeline to generate on the fly a tessellation of the limit surface with a chosen Level of Details.
International audienceLes Systèmes Itérés de Fonctions (IFS) sont un outil standard pour la génér... more International audienceLes Systèmes Itérés de Fonctions (IFS) sont un outil standard pour la génération de formes fractales. Les IFS controlés (CIFS) en sont une extension pour la création de formes fractales à dessein industriel. Un des avantages de cette approche est la possibilité de représenter des surfaces standards comme les surfaces de Bézier, Splines, et de subdivision. La représentation des surfaces par un unique formalisme facilite leur manipulation et la gestion des interactions comme par exemple la construction de raccords entre deux surfaces de natures différentes. Dans cet article, la formulation des B-Splines Rationnelles Non-Uniformes (NURBS) dans le formalisme des CIFS est présentée. Les NURBS étant l'outil central de la plupart des sytèmes CAO, leur ajout à la liste des "objets IFS" permet d'augmenter l'intérêt de ce formalisme dans le cadre d'applications industrielles. En analysant la récursivité du processus de génération des fonctions d...
Discrétisation directe de la surface limite de Catmull-Clark par Systèmes de Fonctions Itérés
International audienc
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Papers by Christian Gentil