Hollow light guides with very high reflective inner surfaces are novel daylight systems that collect sunlight and skylight available on the roof of buildings transporting it into deep or windowless interiors in building cores. Thus the better utilization of daylight can result in energy savings and wellbeing in these enclosed indoor spaces. An analytical complex solution of a straight tube system was solved in the HOLIGILM method with a user-friendly tool available on the http://www.holigilm.info. An even more difficult light flow transport is to be determined in bended tubes usually placed on sloped roofs where a bend is necessary to adjust the vertical pass through the ceilings. This paper presents the theoretical derivation of the model with its graphical representation and coordinate system respecting backward ray-tracing bend distortions. To imagine the resulting illuminance on the horizontal plane element in the interior, the virtual ray (i.e. luminance in an elementary solid angle) has to pass the ceiling diffuser interface, the inner mirror like tube with a bend, through a roof cupola attachment to the element of the sky and sun light source. Due to this complexity and the lengthy derivation and explanations more practical applications will be published later in a separate contribution. (C) 2010 Elsevier Ltd. All rights reserved.