When the quadric is a paraboloid, the relation between Lagrangian and Legendrian singularities is the most natural one, and the calculations, formulas and proofs are simpler. instance: Such a relation holds for any stereographic projection of a hyperplane to a quasi-revolution quadric in ℝn × ℝ, where ℝn denotes Euclidean space. There is a relation between Lagrangian and Legendrian singularities by stereographic projection to a sphere in Euclidean space that we generalise in several directions, for. We establish the correspondence between Euclidean differential geometry of submanifolds in ℝn and projective differential geometry of submanifolds in ℝn+1 under stereographic projection to revolution quadrics (defined below). We present new results on flattenings, Darboux vertices and twistings for curves in Euclidean space of arbitrary dimension. Then the geometry of the points of curvature zero, usually avoided in books and papers, reveals to be very rich and interesting.įlattening’s definition directly generalises to curves of n-dimensional Euclidean spaces while Darboux vertex’s definition has two possible generalisations: one is called Darboux vertex and the other is called twisting. We show that the solution of this ambiguity enters naturally in the frame of Contact Geometry (and we recall some failed attempts). Then we state local and global theorems on flattenings and Darboux vertices of closed curves, and we describe the bifurcations which may occur in generic 1-parameter families of smooth curves (where flattenings and Darboux vertices have an interaction).Īn important point is the discussion on the ambiguity of the Frenet theory at points of curvature zero - at these points the classical theory does not work (the Frenet trihedral is not defined). We start with the local study of smooth curves of Euclidean 3-space near its flattenings (points at which the osculating plane is stationary) and its Darboux vertices (points at which the instantaneous axis of rotation of the Frenet trihedral is stationary). Smooth curves of Euclidean 3-dimensional space and some generalisations for curves of n-dimensional space. We present new theorems on local and global differential geometry of Our results give a formula for calculating the vertices of a curve in ℝn and may be applied to calculate and study umbilic points of surfaces in ℝ 3.

Using the classical theory of poles, polars and polar duality, we construct the natural isomorphism between the front of the Lagrange submanifold of the normal map (considered as a subvariety in J0(ℝn) = ℝn × ℝ) and the front of the Legendre submanifold of the tangential map (considered as a subvariety in the space (ℝn+1)v of affine n-dimensional subspaces in ℝn+1). There is a relation between Lagrangian and Legendrian singularities by stereographic projection to a sphere in Euclidean space that we generalise in several directions, for instance: Such a relation holds for any stereographic projection of a hyperplane to a quasi-revolution quadric in ℝn × ℝ, where ℝn denotes Euclidean space.

Projection sur un hyperplan affine