Rémi Langevin - Academia.edu (original) (raw)

Papers by Rémi Langevin

Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential... more Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential geometry, the simplest after planes and spheres. We prove here that, given three oriented contact conditions, there is in general no Dupin cyclide satisfying them, but if the contact conditions belongs to a codimension one subset, then there is a one-parameter family of solutions, which are all tangent along a curve determined by the three contact conditions.

The set of osculating circles of a given curve in S 3 forms a lightlike curve in the set of orien... more The set of osculating circles of a given curve in S 3 forms a lightlike curve in the set of oriented circles in S 3. We show that its " 1 2-dimensional measure" with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.

We give a direct proof that a codimension–one, C²-foliation F with non-zero Godbillon–Vey class G... more We give a direct proof that a codimension–one, C²-foliation F with non-zero Godbillon–Vey class GV (F) ∈ H³(M) has a hyperbolic resilient leaf. Our approach is based on methods of C¹-dynamics, and does not use the classification theory of C²-foliations. We first prove that for a codimension–one C¹-foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points E(F) has positive Lebesgue measure. We then prove that if E(F) has positive measure for a C¹-foliation F, then F must have a hyperbolic resilient leaf and hence its geometric entropy must be positive. For a C²-foliation, GV (F) non-zero implies the Godbillon measure is also non-zero, and the result follows. These results apply for both the case when M is compact,

The set of osculating circles of a given curve in S forms a curve in the set of oriented circles ... more The set of osculating circles of a given curve in S forms a curve in the set of oriented circles in S. We show that its “ 1 2 -dimensional measure” with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.

Analyser et transcrire une langue de signes est une tache difficile puisque ce mode d'express... more Analyser et transcrire une langue de signes est une tache difficile puisque ce mode d'expression, mouvement des mains dans un espace situe pres du corps, complete par des attitudes et des expressions faciales, est a priori moins sequentiel que la parole. Le travail des AS vise a completer ces nombreuses tentatives anterieures, et s'appuie en particulier sur le dictionnaire de Stoke. En analysant le mouvement d'un repere attache a une main comme le mouvement d'un point dansR 3 × SO(3) ils parviennent a discretiser de maniere naturelle ces gestes les plus frequents de la [LSCB] ou LIBRAS, Brazilian Cities Sign Language. Ce travail permet un premier classement de type alphabetique des signes de la LSCB et donc la constitution d'un premier dictionnaire LSCB → Portugais par L. Ferreira Brito et son equipe

Topology and its Applications, 2018

We renormalize, using suitable lenses, small domains of a singular holomorphic foliation of C 2 w... more We renormalize, using suitable lenses, small domains of a singular holomorphic foliation of C 2 where the curvature is concentrated. At a proper scale, the leaves are almost translates of a graph that we will call profile. When the leaves of the foliations are levels f = λ , where f is a polynomial in 2 variables, this graph is polynomial. Finally we will indicate how our methods may be adapted to study levels of polynomials and 1-forms in C 3 .

Several methods of subdivision exist to build parabola arcs or circle arcs in the usual Euclidean... more Several methods of subdivision exist to build parabola arcs or circle arcs in the usual Euclidean affine plane. Using a compass and a ruler, it is possible to construct, from three weighted points, circles arcs in the affine space without projective considerations. This construction is based on rational quadratic Bezier curve properties. However, when the conic is an ellipse or a hyperbola, the weight computation is relatively hard. As the equation of a conic is qaff(x,y)=1\qaff(x,y)=1qaff(x,y)=1, where qaff\qaffqaff is a quadratic form, one can use the pseudo-metric associed to qaff\qaffqaff in the affine plane and then, the conic geometry is also handled as an Euclidean circle. At each step of the iterative algorithm, the constructed point belongs to a principal perpendicular bissector of the control polyhedron and then, our construction is regular. Moreover, we can pass through the point at infinity when the bounds do not belong to the same branch of the hyperbola, using massic points defined by J.C. Fiorot: we...

Plusieurs methodes de subdivision existent pour construire des arcs de paraboles ou de cercles da... more Plusieurs methodes de subdivision existent pour construire des arcs de paraboles ou de cercles dans le plan affine euclidien usuel. Il est possible de construire des arcs de cercles a la regle et au compas, en restant dans l’espace affine en utilisant trois points ponderes et sans utiliser le concept de geometrie projective. Cette construction s’appuie sur les proprietes des courbes de Bezier rationnelles quadratiques. Cependant, lorsque la conique est un arc d’ellipse ou d’hyperbole, le calcul du poids est relativement complique. Comme l’equation de la conique est Q(x,y) = 1, pour simplifier ce probleme, nous munissons le plan affine de la forme bilineaire symetrique definie Q qui permet de manipuler la conique comme un cercle unitaire : les methodes usuelles, connues dans le cas des cercles euclidiens, peuvent etre alors adaptees. De plus, notre construction est reguliere dans le sens ou, a chaque etape, le point construit sur la conique appartient a la mediatrice principale du tr...

Canal surfaces defined as envelopes of 1-parameter families of spheres, can be characterized by t... more Canal surfaces defined as envelopes of 1-parameter families of spheres, can be characterized by the vanishing of one of the conformal principal curvatures. We distinguish special canals which are characterized by the fact that the non-vanishing conformal principal curvature is constant along the characteristic circles and show that they are conformally equivalent to either surfaces of revolution, or to cones over plane curves, or to cylinders over plane curves, so they are isothermic.

Contemporary Mathematics, 2009

Commentarii Mathematici Helvetici, 1996

The total curvature of compact hypersurfaces M of R" (SM IKI) is related to the topology of M and... more The total curvature of compact hypersurfaces M of R" (SM IKI) is related to the topology of M and to the manner in which M is embedded in R n. K is the Gauss-Kronecker curvature of M, i.e., the determinant of the second fundamental form. For curves C in R 3, the theorems of Fenchet and Fary-Milnor, state the total curvature of C is at least 2zt (with equality precisely for convex planar curves) and if C is knotted in R 3 then 5c Ikt > 4Jr, [Fe], [Fa], [Mi], [M2]. Chern and Lashof observed the total curvature of M* c R" is

Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential... more Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential geometry, the simplest after planes and spheres. We prove here that, given three oriented contact conditions, there is in general no Dupin cyclide satisfying them, but if the contact conditions belongs to a codimension one subset, then there is a one-parameter family of solutions, which are all tangent along a curve determined by the three contact conditions.

In 1872 G. Darboux defined a family of curves on surfaces of R which are preserved by the action ... more In 1872 G. Darboux defined a family of curves on surfaces of R which are preserved by the action of the Möbius group and share many properties with geodesics. Here we characterize these curves under the view point of Lorentz geometry, prove some general properties and make them explicit on simple surfaces, retrieving in particular results of Pell (1900) and Santaló (1941).

We give a new proof of an old theorem by Banchoff and White 1975 that claims that the writhe of a... more We give a new proof of an old theorem by Banchoff and White 1975 that claims that the writhe of a knot is conformally invariant.

Duke Mathematical Journal, 1988

The aim of this paper is to study asymptotic ends of two complete minimal surfaces of finite tota... more The aim of this paper is to study asymptotic ends of two complete minimal surfaces of finite total curvature. Let us suppose that the ends have no ramifica-tion, that is, both surfacesare graphs over theplane orthogonal to the limiting normal. There are two ways to say that one ...

The American Mathematical Monthly

Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential... more Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential geometry, the simplest after planes and spheres. We prove here that, given three oriented contact conditions, there is in general no Dupin cyclide satisfying them, but if the contact conditions belongs to a codimension one subset, then there is a one-parameter family of solutions, which are all tangent along a curve determined by the three contact conditions.

The set of osculating circles of a given curve in S 3 forms a lightlike curve in the set of orien... more The set of osculating circles of a given curve in S 3 forms a lightlike curve in the set of oriented circles in S 3. We show that its " 1 2-dimensional measure" with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.

We give a direct proof that a codimension–one, C²-foliation F with non-zero Godbillon–Vey class G... more We give a direct proof that a codimension–one, C²-foliation F with non-zero Godbillon–Vey class GV (F) ∈ H³(M) has a hyperbolic resilient leaf. Our approach is based on methods of C¹-dynamics, and does not use the classification theory of C²-foliations. We first prove that for a codimension–one C¹-foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points E(F) has positive Lebesgue measure. We then prove that if E(F) has positive measure for a C¹-foliation F, then F must have a hyperbolic resilient leaf and hence its geometric entropy must be positive. For a C²-foliation, GV (F) non-zero implies the Godbillon measure is also non-zero, and the result follows. These results apply for both the case when M is compact,

The set of osculating circles of a given curve in S forms a curve in the set of oriented circles ... more The set of osculating circles of a given curve in S forms a curve in the set of oriented circles in S. We show that its “ 1 2 -dimensional measure” with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.

Analyser et transcrire une langue de signes est une tache difficile puisque ce mode d'express... more Analyser et transcrire une langue de signes est une tache difficile puisque ce mode d'expression, mouvement des mains dans un espace situe pres du corps, complete par des attitudes et des expressions faciales, est a priori moins sequentiel que la parole. Le travail des AS vise a completer ces nombreuses tentatives anterieures, et s'appuie en particulier sur le dictionnaire de Stoke. En analysant le mouvement d'un repere attache a une main comme le mouvement d'un point dansR 3 × SO(3) ils parviennent a discretiser de maniere naturelle ces gestes les plus frequents de la [LSCB] ou LIBRAS, Brazilian Cities Sign Language. Ce travail permet un premier classement de type alphabetique des signes de la LSCB et donc la constitution d'un premier dictionnaire LSCB → Portugais par L. Ferreira Brito et son equipe

Topology and its Applications, 2018

We renormalize, using suitable lenses, small domains of a singular holomorphic foliation of C 2 w... more We renormalize, using suitable lenses, small domains of a singular holomorphic foliation of C 2 where the curvature is concentrated. At a proper scale, the leaves are almost translates of a graph that we will call profile. When the leaves of the foliations are levels f = λ , where f is a polynomial in 2 variables, this graph is polynomial. Finally we will indicate how our methods may be adapted to study levels of polynomials and 1-forms in C 3 .

Several methods of subdivision exist to build parabola arcs or circle arcs in the usual Euclidean... more Several methods of subdivision exist to build parabola arcs or circle arcs in the usual Euclidean affine plane. Using a compass and a ruler, it is possible to construct, from three weighted points, circles arcs in the affine space without projective considerations. This construction is based on rational quadratic Bezier curve properties. However, when the conic is an ellipse or a hyperbola, the weight computation is relatively hard. As the equation of a conic is qaff(x,y)=1\qaff(x,y)=1qaff(x,y)=1, where qaff\qaffqaff is a quadratic form, one can use the pseudo-metric associed to qaff\qaffqaff in the affine plane and then, the conic geometry is also handled as an Euclidean circle. At each step of the iterative algorithm, the constructed point belongs to a principal perpendicular bissector of the control polyhedron and then, our construction is regular. Moreover, we can pass through the point at infinity when the bounds do not belong to the same branch of the hyperbola, using massic points defined by J.C. Fiorot: we...

Plusieurs methodes de subdivision existent pour construire des arcs de paraboles ou de cercles da... more Plusieurs methodes de subdivision existent pour construire des arcs de paraboles ou de cercles dans le plan affine euclidien usuel. Il est possible de construire des arcs de cercles a la regle et au compas, en restant dans l’espace affine en utilisant trois points ponderes et sans utiliser le concept de geometrie projective. Cette construction s’appuie sur les proprietes des courbes de Bezier rationnelles quadratiques. Cependant, lorsque la conique est un arc d’ellipse ou d’hyperbole, le calcul du poids est relativement complique. Comme l’equation de la conique est Q(x,y) = 1, pour simplifier ce probleme, nous munissons le plan affine de la forme bilineaire symetrique definie Q qui permet de manipuler la conique comme un cercle unitaire : les methodes usuelles, connues dans le cas des cercles euclidiens, peuvent etre alors adaptees. De plus, notre construction est reguliere dans le sens ou, a chaque etape, le point construit sur la conique appartient a la mediatrice principale du tr...

Canal surfaces defined as envelopes of 1-parameter families of spheres, can be characterized by t... more Canal surfaces defined as envelopes of 1-parameter families of spheres, can be characterized by the vanishing of one of the conformal principal curvatures. We distinguish special canals which are characterized by the fact that the non-vanishing conformal principal curvature is constant along the characteristic circles and show that they are conformally equivalent to either surfaces of revolution, or to cones over plane curves, or to cylinders over plane curves, so they are isothermic.

Contemporary Mathematics, 2009

Commentarii Mathematici Helvetici, 1996

The total curvature of compact hypersurfaces M of R" (SM IKI) is related to the topology of M and... more The total curvature of compact hypersurfaces M of R" (SM IKI) is related to the topology of M and to the manner in which M is embedded in R n. K is the Gauss-Kronecker curvature of M, i.e., the determinant of the second fundamental form. For curves C in R 3, the theorems of Fenchet and Fary-Milnor, state the total curvature of C is at least 2zt (with equality precisely for convex planar curves) and if C is knotted in R 3 then 5c Ikt > 4Jr, [Fe], [Fa], [Mi], [M2]. Chern and Lashof observed the total curvature of M* c R" is

Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential... more Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential geometry, the simplest after planes and spheres. We prove here that, given three oriented contact conditions, there is in general no Dupin cyclide satisfying them, but if the contact conditions belongs to a codimension one subset, then there is a one-parameter family of solutions, which are all tangent along a curve determined by the three contact conditions.

In 1872 G. Darboux defined a family of curves on surfaces of R which are preserved by the action ... more In 1872 G. Darboux defined a family of curves on surfaces of R which are preserved by the action of the Möbius group and share many properties with geodesics. Here we characterize these curves under the view point of Lorentz geometry, prove some general properties and make them explicit on simple surfaces, retrieving in particular results of Pell (1900) and Santaló (1941).

We give a new proof of an old theorem by Banchoff and White 1975 that claims that the writhe of a... more We give a new proof of an old theorem by Banchoff and White 1975 that claims that the writhe of a knot is conformally invariant.

Duke Mathematical Journal, 1988

The aim of this paper is to study asymptotic ends of two complete minimal surfaces of finite tota... more The aim of this paper is to study asymptotic ends of two complete minimal surfaces of finite total curvature. Let us suppose that the ends have no ramifica-tion, that is, both surfacesare graphs over theplane orthogonal to the limiting normal. There are two ways to say that one ...

The American Mathematical Monthly