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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)
(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 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)