Abstract : A convex G2 Hermite interpolation problem of concentric curvature elements is considered in this paper. It is first proved that there is no spiral arc solution with turning angle less than or equal to $\pi$ and then, that any convex solution admits at least two vertices. The curvature and the evolute profiles of such an interpolant are analyzed. In particular, conditions for the existence of a G2 convex interpolant with prescribed extremal curvatures are given.

Abstract : This paper deals with the motion capture of physical surfaces via a curve acquisition device. This device is a ribbon of sensors, named Ribbon Device, providing tangential measurements, allowing to reconstruct its 3D shape via an existing geometric method. We focus here on the problem of reconstructing animated surfaces, from a finite number of curves running on these surfaces, acquired with the Ribbon Device. This network of spatial curves is organized according a comb structure allowing to adjust these curves with respect to a reference curve, and then to develop a global C1 reconstruction method based on the mesh of ribbon curves together with interpolating transversal curves. Precisely, at each time position the surface is computed from the previous step by an updating process.

Abstract : The fact that the Darboux frame is rotation--minimizing along lines of curvature of a smooth surface is invoked to construct rational surface patches whose boundary curves are lines of curvature. For given patch corner points and associated frames defining the surface normals and principal directions, the patch boundaries are constructed as quintic RRMF curves, i.e., spatial Pythagorean--hodograph (PH) curves that possess rational rotation--minimizing frames. The interior of the patch is then defined as a Coons interpolant, matching the boundary curves and their associated rotation--minimizing frames as surface Darboux frames. The surface patches are compatible with the standard rational B\'ezier/B--spline representations, and $G^1$ continuity between adjacent patches is easily achieved. Such patches are advantageous in surface design with more precise control over the surface curvature properties.

Abstract : Given three regular space curves ${\bf r}_1(t)$, ${\bf r}_2(t)$, ${\bf r}_3(t)$ for $t\in[\,0,1\,]$ that define a curvilinear triangle, we consider the problem of constructing a triangular surface patch ${\bf R}(u_1,u_2,u_3)$ bounded by these three curves, such that they are geodesics of the constructed surface. Results from a prior study \citep{Farouki09} concerned with tensor--product patches are adapted to identify constraints on the given curves for the existence of such geodesic--bounded triangular surface patches. For curves satisfying these conditions, the patch is constructed by means of a cubically--blended triangular Coons interpolation scheme. A formulation of thin--plate spline energy in terms of barycentric coordinates with respect to a general domain triangle is also derived, and used to optimize the smoothness of the geodesic--bounded triangular surface patches.

Abstract : For a given polynomial $F(t)= \sum_{i=0}^{n} p_i B_i^n(t)$, expressed in the Bernstein basis over an interval $[a,b]$, we prove that the number of real roots of $F(t)$ in $[a,b]$, counting multiplicities, does not exceed the sum of the number of real roots in $[a,b]$ of the polynomial $G(t)= \sum_{i=k}^l p_i B_{i-k}^{l-k}(t)$ (counting multiplicities) with the number of sign changes in the two sequences $(p_0,...,p_k)$ and $(p_l,...,p_n)$ for any value $k,l$ with $0\leq k \leq l \leq n$. As a by product of this result, we give new refinements of the classical variation diminishing property of B\'ezier curves.

Abstract : Given four polynomial or rational B\'ezier curves defining a curvilinear rectangle, we consider the problem of constructing polynomial or rational tensor--product B\'ezier patches bounded by these curves, such that they are geodesics of the constructed surface. The existence conditions and interpolation scheme, developed in a general context in earlier studies, are adapted herein to ensure that the geodesic--bounded surface patches are compatible with the usual polynomial/rational representation schemes of CAD systems. Precise conditions for four B\'ezier curves to constitute geodesic boundaries of a polynomial or rational surface patch are identified, and an interpolation scheme for the construction of such surfaces is presented when these conditions are satisfied. The method is illustrated with several computed examples.

Abstract : Given two pairs of regular space curves ${\bf r}_1(u)$, ${\bf r}_3(u)$ and ${\bf r}_2(v)$, ${\bf r}_4(v)$ that define a curvilinear rectangle, we consider the problem of constructing a $C^2$ surface patch ${\bf R} (u,v)$ for which these four boundary curves correspond to geodesics of the surface. The possibility of constructing such a surface patch is shown to depend on the given boundary curves satisfying two types of consistency constraints. The first constraint is global in nature, and is concerned with compatibility of the variation of the principal normals along the four curves with the normal to an oriented surface. The second constraint is a local differential condition, relating the curvatures and torsions of the curves meeting at each of the four patch corners to the angle between those curves. For curves satisfying these constraints, the surface patch is constructed using a bicubically--blended Coons interpolation process.

Abstract : A constructive geometric approach to rational ovals and rosettes of constant width formed by piecewise rational PH curves is presented. We propose two main constructions. The first construction, models with rational PH curves of algebraic class $3$ (T-quartics) and is based on the fact that T-quartics are exactly the involutes of T-cubic curves. The second construction, models with rational PH curves of algebraic class $m > 4$ and is based on the dual control structure of offsets of rational PH curves.

Abstract : This paper is concerned with reconstruction of numerical or real surfaces based on the knowledge of some geodesic curves on the surface. So, considering two regular 3D-curves $f_0(t)$ and $f_1(t)$, our purpose is to construct a surface which interpolates these two curves in such a way that these two curves are geodesics on this surface. This will be accomplished using Hermite interpolation. For a real surface, it will be shown that geodesics can be acquired using a ribbon of micro-sensors.

Abstract : This paper presents a novel method for recontructing surfaces relying only on tangential data which are provided by embedded sensors. The reconstruction process is based on the knowledge of the distribution of the sensors, which are organized as a square mesh, and on the associated tangential orientation measurements without any information about their positioning in space so that this problem cannot be solved by envelope methods. We provide methods for planar and spatial curves, then extend them for surfaces. We validate these methods proving their convergence and by the analysis of the results obtained in a physical point of view. Finally, we show the implementation of our work in a real time prototype.

Abstract : This paper presents a novel method for reconstructing curves relying on tangential data which are provided by embedded sensors. The reconstruction process is based on the knowledge of the distribution of the sensors along the curve, represented by a ribbon, and on the associated tangential orientation measurements without any information about their positioning in space, so that this problem is not an envelope problem. We first show how we can obtain these data from sensors and the prototypes we have created. Then we provide methods for planar curves, then for spatial curves and we analyze results with physical sense and convergence in order to validate these methods. Finally, we show some results from both simulated data and real data.

Mat234 (Licence 2)

Résumé cours analyse (pdf)

Résumé cours algèbre (pdf)

Fiches TD (pdf)
-- Corrigés TD analyse (pdf)
-- Corrigés TD algèbre (pdf)

Annales

CC1 novembre 2009 (pdf) - Corrigé (pdf)

CC2 décembre 2009 (pdf) - Corrigé (pdf)

Examen1 décembre 2009 (pdf)

Examen2 juin 2010 (pdf)

CC1 novembre 2010 (pdf) - Corrigé (pdf)

CC2 décembre 2010 (pdf)

Examen1 janvier 2011 (pdf)

Examen2 juin 2011 (pdf)

CC1 novembre 2011 (pdf) - Corrigé (pdf)

CC2 décembre 2011 (pdf) - Corrigé (pdf)

Examen1 janvier 2012 (pdf) - Corrigé (pdf)

Examen2 juin 2012 (pdf)

CC1 octobre 2012 (pdf)

CC2 décembre 2012 (pdf) - Corrigé (pdf)

Examen1 janvier 2013 (pdf) - Corrigé (pdf)

Géométrie Appliquée (M1 MAI)

TP1 : courbe Bézier - subdivision (pdf)

TP2 : interpolation Bézier (pdf)

TP3 : offset et evolute/involute (pdf)

TP4 : principe d'exclusion (pdf)

TP5 : spline cubique (pdf)

TP6 : directions principales - Dupin (pdf)

TP7 : deBoor-Cox (pdf)

TP8 : insertion noeuds et subdivision spline (pdf)

TP9 : spline périodique (pdf)

TP10 : Bézier rationnelle (pdf)

TP11 : courbe Nurbs (pdf)

TP12 : surface Bézier tensorielle(pdf)

TP13 : surface Bézier tensorielle rationnelle (pdf)

TP14 : surface Nurbs (pdf)