  We present the list of unavoidable local phenomena (transitions) occurring on
the configuration of the parabolic and flecnodal curves of evolving smooth
surfaces in R^3 (or RP^3). We also present the list of transitions occurring on
the curve of inflections of the solutions of evolving implicit differential
equations (IDE). Our results are based on the properties of the contours of
surfaces (in a contact 3-space) for projections all whose fibres are
Legendrian. Keywords: Surface, flecnodal curve, contact geometry, implicit
differential equations.
