Our aim in this paper is to define principal and characteristic directions at points on a smooth 2-dimensional surface in the Euclidean space R 4 in such a way that their equations together with that of the asymptotic directions behave in the same way as the triple formed by their counterpart on smooth surfaces in the Euclidean space R 3. The definitions we… (More)
In this paper we consider singularities of orthogonal projections of piecewise-smooth surfaces on planes. For a generic smooth surface, the apparent contour (outline, profile) of the surface associated with a projection onto a plane is the set of critical values of the projection. The singularities of apparent contours of smooth surfaces have been studied… (More)
We obtain the topological configurations of the lines of curvature, the asymptotic and characteristic curves on a cross-cap, in the domain of a parametrisation of this surface as well as on the surface itself.
We obtain in this paper topological models of binary differential equation at local codimension 2 singularities where all the coefficients of the equation vanish at the singular points. We also study the bifurcations of these singularities when the equation is deformed in a generic 2-parameter families of equations.
(Communicated by Aim Sciences) Dedicated to Carlos Gutierrez and Marco Antonio Teixeira on the occasion of their 60th birthdays. Abstract. We study geometric properties of the integral curves of an implicit differential equation in a neighbourhood of a codimension ≤ 1 singularity. We also deal with the way these singularities bifurcate in generic families… (More)
We prove that any closed and convex surface in the Minkowski 3-space of class C 3 has at least two umbilic points. This shows that the Carathéodory conjecture for surfaces in the Euclidean 3-space is true for surfaces in the Minkowski 3-space.
We define and study in this paper families of conjugate and reflected curve congruences associated to a self-adjoint operator A on a smooth and oriented surface M endowed with a Lorentzian metric g. These families trace parts of the pencil joining the equations of the A-asymptotic and the A-principal curves, and the pencil joining the A-characteristic and… (More)
We survey in this paper results on a particular set of Implicit Differential Equations (IDEs) on smooth surfaces, called Binary/Quadratic Differential Equations (BDEs). These equations define at most two solution curves at each point on the surface, resulting in a pair of foliations in some region of the surface. BDEs appear naturally in differential… (More)
We define in this paper the asymptotic, characteristic and principal directions associated to the de Sitter Gauss map on a smooth timelike surface M in the de Sitter space S 3 1. We study their properties and determine the local topological configurations of their integral curves. These curves form pairs of foliations on some regions of M and are defined in… (More)