Source genericity of caustics by reflexion in R 3

Abstract

Let M (a mirror) be a compact ovaloid in R n+1 , and let s (a light source) be a point in the ambient space. Rays of light emanating from s are reflected at M , giving rise to a caustic C s , the envelope of the family of reflected rays. The corresponding wavefront W s (the orthotomic) is the locus of reflexions of s in the affine tangent planes to M , and is smooth if and only if s lies inside M , which is henceforward assumed to be the case. When W s is generic, i. e. the family of distance-squared functions on W s is transverse to the canonical stratification of the jet-space, a theorem of Looijenga tells us that the caustic C s can be described (locally) by a finite number of models. A fundamental question is that of source genericity : can we force W s to be generic by small deformations of s ? For n = 1, when the mirror is a plane oval, the authors have shown that this is the ease provided one excludes certain very special types of mirror. In the present paper we consider the local aspect of source genericity for n = 2; more explicitly, we consider transversality to strata of corank 2, and strata of corank 1 of co-dimension ≼ 4, i. e. A 1 , A 2 , A 3 , A 4 . The key idea behind all the work is that studying the contact of a sphere with W s is essentially the same as studying the contact of a quadric of revolution (namely the anti-orthotomic of the sphere with respect to the source) with the mirror M . In this way the problem on W s (which varies with s ) is reduced to a problem on M which is technically easier to handle. The positive results obtained are as follows. Theorem. (i) Suppose that the set of umbilics on M has Lebesgue measure zero in M. Then, for almost all positions of the source s inside M the family of distance-squared functions on the orthotomic Ws is transverse to all strata of corank 2. (ii) Assume that M is analytic. Suppose that the set of non-transverse A≽4 points on Mis subanalytic of codimension ≽ 1 in M, and that there are only finitely many umbilics, none of which are ‘bad' (in a technical sense). Then, for almost all positions of the source s inside M the family of distance-squared functions on the orthotomic W s is transverse to the strata A k with K ≼ 4 . More interesting perhaps is the fact that we can obtain (non-trivial) counterexamples to source genericity. Thus mirrors with ‘bad’ umbilics provide counterexamples, as do surfaces of revolution.