SMS freeforms for illumination

Abstract

SMS 3D (simultaneous multiple surfaces in their three-dimensional version) is a well-known design method comprising two freeform surfaces that allow the perfect coupling of two wavefronts with another two. The design algorithm provides a collection of line pairs on both surfaces (called SMS spines), whose three-dimensional shape seems arbitrary at first sight. This paper shows that the shapes of the spines are partially governed by applying the étendue conservation theorem to the biparametric bundle of rays linking the paired spines, which is one lesser known éten-due invariants found by Poincaré. The resulting formulae for the spines in three-dimensional space happen to coincide with the conventional étendue formulas of two-dimensional geometry, like for instance, the Hottel formula.

Análise gemológica e de inclusões em berilos da região de Pedra da Onça - Santa Tereza- ES

“…As características do depósito são de um pegmatito intermediário, zonado e pouco diferenciado quimicamente 1,13,14,15,16,19,20,21…”