Planar Sections of a Surface Close to an Umbilic

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