Regiomontanus angle maximization for catadioptric sensors with paraboloidal mirrors

Dioptric cameras with conventional perspective projection have well established analytical properties. However, they suffer from perspective distortions and only have a limited field of view. Catadioptric cameras offer panoramic imaging. Their extensive field of view together with projection specific image analysis, can simplify many computer vision tasks. Several properties of catadioptric projection for geometric primitives such as points and lines have been addressed and have also been used for calibration. However, higher order geometric properties are yet to be investigated. Such analysis is complicated by the specifics of the warping of the scene by catadioptric projection. One such property, that is the subject of this work, is the Regiomontanus angle maximization relative to the effective viewpoint of the sensor. This work considers catadioptric sensors with paraboloidal mirrors, that is, paracatadioptric sensors. Analytical ray tracing of a simplified 1D world object gives its projection in the image and an expression for its length. The optimization of the length of the projection results in a third degree equation for the Regiomontanus distance that can be solved explicitly. The Khayyam geometric solution of this equation provides the Regiomontanus distance of maximum subtended projection for these cameras. Applications of these results in various contexts are presented and discussed.