This paper describes a novel projection model based on the so-called unified projection model. The new model applies to catadioptric systems and wide-angle fish-eye cameras, it does not require additional mapping to model distortions, and it takes just two projection parameters more than a simple pinhole model to represent radial distortion (one parameter more than the unified model). Here we provide a study of different mathematical aspects of the model, its application limits, and explicit closed-form inversion. The latter allows to apply all the notions of epipolar geometry with no difficulties. Also we introduce a concept of projection surface, which is a useful notion to study and compare different projection models with radial distortion. Using developed software, several different lenses were calibrated using the proposed model, and in all cases sub-pixel precision was achieved.