2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163255It has been known that, keeping the outer boundary coordinates intact before and after a coordinate transformation is a sufficient condition for obtaining a reflectionless transformation medium. Here we prove that it is also a necessary condition for reflectionless transformation media in an isotropic and homogenous background. Our analytical results show that the outer boundary coordinates of a reflectionless transformation medium must be the same as the original coordinates with a combination of rotation and displacement, which is equivalent to situation that the boundary coordinates are kept intact before and after transformation.9 pages, 4 figuresOpticsClassical Physicstext