About metric and angular dependencies of spatial straight line notches and their use in engineering and geodetic works

In applied geodesy tasks, it may be necessary to determine spatial angles. When bringing a 3D design of buildings and structures to the field with the help of an electronic total station (ES), it is important to verify the spatial angles between different elements of building structures such as roof overlaps, inclined anchors, and more, using the characteristic points' spatial coordinates. Modern geodetic instruments provide sufficiently high measurement accuracy (up to 1" and 1 mm, respectively). However, measuring the required angles with surveying instruments is not always possible for various reasons. First of all, it is impossible to place the device at the vertex of an angle if the location is not accessible. This paper develops a method for determining a spatial angle whose vertex is not available for measurement. Methods and results. To achieve this goal, we consider one of the options for its determination through the application of the cosine theorem with preliminary measurement or calculation of adjacent sides and vertical angles. This article also presents an algorithm for solving the problem with an estimation of the accuracy of establishing the required parameters. The basic formulas for determining the angles of a spatial triangle with an estimate of their accuracy are proposed. The paper studies the influence of the linear measurement values of the lengths of the sides on the values of the angles of a spatial triangle with the corresponding accuracy assessment. In particular, the root mean square errors of angle calculation were determined based on these calculations and mathematical modeling, namely, the ratio of the sides of the triangle. Through indirect measurements of the tower crane boom and roof spire, the spatial angle values were determined. The inclination of the crane boom to the base resulted in α=910.712±51", while the angle of the roof spire was α=150.109±35". Scientific novelty and practical significance. On the basis of the proposed methodology and numerical experiments, spatial angles were determined and their a priori accuracy was analyzed. This confirms the influence of linear measurements of side lengths on the values of spatial angles. The obtained results make it possible to apply the proposed method in engineering and geodetic works using BIM technologies in 3D space. This method can be used in the application software of electronic total station manufacturers to determine spatial angles in space when solving engineering problems.