As an example the normal sections in the vertex of the elliptic paraboloid "x^2/r1 + y^2/r2 = 2*z" are regarded. The angle distance between the normal sections is 15° each. The envelope of the circles is an algebraic surface of fourth order with the equation "(x^2 + y^2 + z^2)(x^2/r_1 + y^2/r_2) - 2z(x^2 + y^2) = 0". The envelope is homeomorphic to the projective plane and in this topological context often refered to as cross-cap.

As an example for the elliptic surface point, the vertex P of the elliptic paraboloid \frac{x²}{r_1}+ \frac{y²}{r_2} = 2z; (r_1 = 10, r_2 = 5) was chosen. In general: to every regular surface point P a paraboloid can be chosen, so that it osculates with the vertex of the surface, which means it the surface coincides with the curvature in P. The so-called vertex paraboloid can by elliptic or hyperbolic, or a parabolic cylinder.