Caustics in extended euclidean space

• Published in: SIAM journal on mathematical analysis 1984, Vol.15 (1), p.50-68
• Main Authors: BRUCE, J. W, GIBLIN, P. J, GIBSON, C. G
• Format: Article
• Language: English
• Subjects: Applied mathematics; Euclidean space; Exact sciences and technology; Function theory, analysis; Light; Mathematical methods in physics; Mirrors; Physics
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/0515002

This paper is one of a series in which the authors investigate genericity properties of caustics by reflexion from smooth mirrors. Here, the mirror is a smooth surface in $\mathbb{R}^3 $, and two related problems are considered: (1) what is the form of the caustic at infinity (the caustic in the "far field"), generically, with a finite light source; (2) what is the form of the caustic generically when the light source is at infinity? For (1) both "mirror genericity," where deformations of the mirror are allowed, and "source genericity," where only the source may be moved, are considered. With some assumptions on $M$ it is shown that the caustic can be made generic in either of these two ways. (In the latter case, only a local result is proved.) For (2), only mirror genericity is considered, and it is shown that there is an obstruction to proving a result of this kind, caused by an inherent lack of genericity in wavefronts arising from parallel light reflected from a mirror in $\mathbb{R}^3 $. It is proved, however, that for most mirrors in $\mathbb{R}^3 $ with parallel incident light, the part of the caustic lying in a compact region is generic.