Planar Sections of a Surface Close to an Umbilic

We study the intersection between a smooth algebraic surface with an umbilic point and a plane parallel and close to the tangent plane at the umbilic. The problem has its origin in the study of isophote (equal illumination) curves in a 2-dimensional image. In particular, we study the circles which have exceptional tangency to this intersection curve: ordinary tangency at one point and osculating at another; ordinary tangency at three points; and 4-point tangency at a vertex. The centres of circles having ordinary tangency at two points trace out a curve whose closure is the symmetry set of the intersection curve, and the exceptional circles above give respectively cusps, triple crossings and endpoints of this set. We analyse the curves traced out by the contact points and centres of the exceptional circles as the plane approaches the tangent plane at the umbilic. We also briefly discuss the global structure of the symmetry set by means of a typical example.