A general tool for the calculation of umbilical singularities using Thom's singular set Σ2

From the mathematical viewpoint, a congruence of rays may be modelled by some mapping. In this frame, the caustic corresponds to the points where the rank of the mapping decreases by at least one unit. The umbilical singularities are the special points where the rank decreases by two units and they constitute, by definition, the Thom singular set Σ2. We propose to exploit this mathematical property to calculate directly the umbilical singularities, without any calculation of the caustic surface. The method is applied in two explicit examples. In the first example we consider a wavefront and we show that the equations of the set Σ2 lead to the classical result on the equality of the principal curvatures on the wavefront. In the second example we use the Σ2 method for the localization of umbilics produced by a paraboloidal reflector. For comparison, we present also the same calculation by starting from the equation of the caustic