SynopsisIn this paper we give the local classification of solution curves of bivalued direction fields determined by the equationwhere a and b are smooth functions which we suppose vanish at 0 ∈ ℝ2. Such fields arise on surfaces in Euclidean space, near umbilics, as the principal direction fields, and also in applications of singularity theory to the structure of flow fields and monochromatic-electromagnetic radiation. We give a classification up to homeomorphism (there are three types) but the methods furnish much additional information concerning the fields, via a crucial blowing-up construction.

Let M (the mirror) be a plane oval (a smooth curve without inflexions), and let sεℝ2\M be the light source. Rays of light emanating from s are reflected by M, and the caustic by reflexion of M relative to s is the envelope of these reflected rays. In this article we suppose that M is generic (the precise assumption is stated later) and that s moves along a smooth curve in the plane; we are then able to describe how the local structure of the caustic changes. In order to state the result we recall a few facts from [3].

Let M be a plane oval (a smooth curve without inflexions). In this note we show that a generic such M (where the precise assumptions will be stated later) has to have at least one sextactic point, that is a point p where the unique conic touching M at p with at least 5-point contact actually has 6-point contact. This existence problem came into prominence whilst [2] was being written. It was hoped to use the existence of sextactic points to show that the Morse transition on a 1-parameter family of focoids with signature 0 or 2 could not occur. The problem proved to be remarkably stubborn, however. Indeed, the geometric interpretation of sextactic points as given in § 3 was totally unexpected.