Investigation of formulas determination of a point’s plane coordinates by the invers linear-angular resection

The aim. The study of formulas determination of the point coordinates by the inverse linear-angular intersection method. Previously, we investigated the possibility of using electronic total stations to control the geometric parameters of industrial buildings. The applied application of electronic total stations for high-precision measurements has been investigated as well. [Vivat, 2018]. The formula for optimal use of the device with certain accuracy characteristics relative to the measured basis is analytically proved and derived [Litynskyi, 2014]. Measurements on the basis of the II category are performed and theoretical calculations are confirmed. The possibility of achieving high accuracy in determining the segment by the method of linear-angular measurements is shown [Litynsky, 2015]. The influence of the angle value on the accuracy of determining the coordinates by the sine theorem is investigated and the possibility of optimizing the determination of coordinates by the method of inverse linear-angular serif by the formulas of cosines and sines is investigated [Litynskyi, 2019]. Method. Establishing a mathematical interconnection between measured values (distances and angles) with the required (flat coordinates of a point), differentiation and finding the minima of functions. Results.There were five formulas selected, of which six combinations had been created to calculate the increments of coordinates and to estimate their accuracy. Numerical experiments show that neither method has a significant advantage, which is supported by the results presented in the graphs and tables. It is worth noting one feature of the second method - in which it is possible to determine the increments of coordinates with an accuracy that exceeds the accuracy of measuring the sides. The possibility of optimizing the coordinate increments determination due to the choice of calculation formulas is considered. The possibility of increasing the accuracy of determination of the coordinates increments using different calculation formulas is researched. Consequently, it is suggested to optimize the choice of calculation formulas depending on the position of the desired point. The results of these studies can be used to create electronic total station or laser tracker application software in order to improve the accuracy of coordinate determination.