THE COMBINED GEOMETRICAL MODEL IN THE OPTIMIZING APPROACH TO DETERMINATION THE PARAMETERS OF AN INACCESSIBLE POINT OF AN OBJECT

Abstract

In the present research the combined geometrical model and the optimizing approach to the determination of the parameters of an inaccessible point of an object is developed. The common issues are revealed and essential steps of their resolution are identified. The essence of the problem is that the unavoidable contradiction between a need of obtaining the exact value of the determined parameter and an error involved in any measurement. The purpose of the present work is to develop in a complex the combined threedimensional geometrical and analytical models of the determination of the minimum domain (area, vicinity) of values of the parameters of an inaccessible point of an object. It is achieved in two steps: 1. Development of a combined three-dimensional geometrical model with crossed directional rays for contactless determination of the coordinates of an inaccessible point of an object under a given arrangement of the geodetic equipment. 2. Development of an optimizing analytical model for the determination of domain (area, vicinity) of values of parameters of an inaccessible point of an object according to the developed three-dimensional geometrical model with crossed directional rays. In the proposed optimizing approach, the combined three-dimensional geometrical model with crossed directional rays for the determination of coordinates of the inaccessible points of an object is developed. It is discussed that point C, coordinated of which to be determined, locates in domain [CDM, CEM] of the minimum distance ρmin between crossed directional rays. The optimizing problem of the determination of coordinates of an inaccessible point of an object in space is reduced to a problem of the determination of the minimum distance between two crossed directional rays. It is shown that the problem has a unique solution if the directional rays are not parallel. It is discussed that mathematical finding of an extremum of function of the distance between two directional rays, and the discussed minimum, has real geometrical interpretation. It’s known from the theory of function of multiple variables that function f(tCD, tCE) reaches its extremum ρmin when its partial derivatives by each variable are equal to zero. Therefore, the system of the differential equations solved. The required point C(xC, yC, zC) can be located, for example, in the middle of the minimum segment [CDM, CEM]. The proposed theoretical approach is verified using real experimental data at restoration of the Spaso-Preobrazhenskiy cathedral in the city of Odessa, Ukraine. Coordinates of the highest point of the colon of a pilaster С and points С' level of the earth concerning a horizontal plane with zero directional rays were determined.