The author establishes the correspondence between Euclidean differential geometry of submanifolds in \(\mathbb R^n\) and projective differential geometry of submanifolds in \(\mathbb R^{n+1}\) under the stereographic projection to revolution quadrics. V.~D.~Sedykh showed that the Lagrange singularities of the normal map associated with a submanifold in Euclidean space correspond to the Legendre singularities of the tangential map associated with its image under the stereographic projection to the sphere. In this paper, Sedykh's theorem is generalized in several directions. It is shown that the correspondence between Lagrange and Legendre singularities can be formulated more explicitly. Namely, the construction of the natural isomorphism is given between the front of the Lagrange submanifold of the normal map (considered as a subvariety in \(J^0(\mathbb R^n) = \mathbb R^n\times\mathbb R)\) and the front of the Legendre submanifold of the tangential map (considered as a subvariety in the space \((\mathbb R^{n+1})^\vee\) of affine \(n\)-dimensional subspaces in \(\mathbb R^{n+1}\)). This isomorphism is given by the polar duality map or polar reciprocal map with respect to the sphere. As a consequence, it is shown that the Lagrangian normal map of a submanifold \(\Gamma\) in \(\mathbb R^n\times \{b\}\) can be constructed as the composition of the Legendrian tangential map of the manifold \(\rho(\Gamma)\), the polar duality map and the `vertical' projection of \(\mathbb R^{n+1}= \mathbb R^n \times\mathbb R\) into \(\mathbb R^n\times \{b\}\) from the pole \(N\) of the stereographic projection. In particular, the projection from \(N\) of the `cuspidal edge' of the polar dual front of \(\rho(\Gamma)\) into \(\mathbb R^n\times \{b\}\) is the caustic of \(\Gamma\) by the normal map. It is also shown that a similar correspondence extends to the case when the sphere used in the definition of the stereographic projection and of the polar duality is replaced by any quadric of revolution being ellipsoid, paraboloid or even a two-sheeted hyperboloid in \(\mathbb R^n\times \mathbb R\). Moreover, the correspondence is most natural and the formulas are the most simple in the case when the quadric of revolution is a paraboloid. In particular, the stereographic projection of the hyperplane \(\mathbb R^n\times\{0\}\subset \mathbb R^n\times \mathbb R\) into the revolution paraboloid given by the equation \(z = \frac{1}{2}(x^2_1+\dots+x^2_n)\) is just the vertical projection \[ \rho: (x_1,\dots,x_n,0)\mapsto (x_1,\dots,x_n,\tfrac{1}{2}(x_1^2 + \dots+x^2_n)). \] As a consequence of these theorems, the author obtains the following corollary: The vertices of a curve \(\gamma: \mathbb R\to \mathbb R^n\times \{0\}\), \(g(t) = (\gamma_1(t),\dots,\gamma_n(t), 0)\), are sent under the projection \(\rho\) onto the flattenings of the curve \(\widetilde\gamma(t) = (\gamma_1(t),\dots,\gamma_n(t),\frac{1}{2}[\gamma_1^2(t) + \dots + \gamma_n^2(t)])\). These results give a formula for calculating the vertices of a curve in \(\mathbb R^n\) and may be applied to calculate and study umbilic points of surfaces in \(\mathbb R^3\).