Equivalence of catadioptric projections and mappings of the sphere

In this paper we demonstrate that all single viewpoint catadioptric projections are equivalent to the composition Of central projection to the sphere followed by a point projection from the sphere to an image plane. Special cases of this equivalence are parabolic projection, for which the second map is a stereographic projection, and perspective projection, for which the second map is central projection. We also show that two projections are dual by the mapping which takes conics to their foci. Thc foci of line images are points of another, dual, catadiotpric projection; and vice versa, points in the image are foci of lines in the dual projection. They are dual because the mapping preserves incidence relationships. Finally we show some applications of the theory presented above. We present a general algorithm for calibration of a catadioptric system with fines in a single view.