V. G. Andronov, I. A. Klochkov, E. V. Lazareva

For the first time, from the unified position of the set theory the authors gave general mathematical formulation of the primal photogrammetric problem and solved it for different types of monoscopic space images in the specified cartographic projection for any shape of the ground in the region of photography. In this regard the authors offered to use a priori known mean value of the shape of the ground in the region of photography in the capacity of geodesic altitude in conventional collinearity equations. It is shown that calculated plane coordinates will be - for every point of the space image - the coordinates of the reflection of the cross points of the optic axis of the satellite-borne shooting equipment and the virtual mean contour line of the relief onto the local horizon plane in the specified cartographic projection. Then the reflections of these points with calculated coordinates are considered the origins of the accessory reference systems, and the axes of these systems are parallel to the axes of the coordinate systems of the specified cartographic projection. The approach presented in the paper made it possible to get the simple linear equations describing functional relationship between the plane coordinates of the cross points of the optic axis of the satellite-borne shooting equipment with the earth surface and the virtual mean contour line of the relief. In contrast to known approaches the proposed method of solution of the primal photogrammetric problem allows to reduce computational cost by an order. Firstly this result is achieved by reduction of the laborious iterative operations of solving of the classical system of collinearity equations to the computational procedure applied to every point of the monoscopic space image only once. Secondly it is achieved by organization of the iterative process for some linear parameter that is connected to the sought-for plane coordinates of the earth surface point by simple functional relation-ship. The transcendental equation obtained in the article was called by the authors the «fundamental equation». This equation has no analogues in known literature, it is compact, has rectilinear contiguous work region, and as well it is a priori exact with known initial estimate. Furthermore, absolute value of the first-order derivative of its right side is always less then one, which ensures the necessary and sufficient condition for convergence of iterative process and uniqueness of the obtained results