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

2023;
: 79-88

1
Department of Cartography and Geospatial Modeling, Institute of Geodesy, Lviv Polytechnic National University
2
Lviv Polytechnic National University
3
Engineering geodesy department of Lviv Polytechnic National University
4
Department of Cartography and Geospatial Modeling of Lviv Polytechnic National University

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.

1. Antonuk, V., Astafev, V., Savchuk, S., Vivat, А., & Shevchenko, Т. (2006). Comprehensive implementation of the method of installing equipment in the design position using modern and traditional geodetic equipment. Geodesy, Cartography, and Aerial Photography, (67), 10-16. (in Ukrainian). https://science.lpnu.ua/istcgcap/all-volumes-and-issues/volume-67-2006/s...
2. Baran, P. (2012). Engineering geodesy. Kyiv: PAT «VIPOL», 2012. P. 618. (in Ukrainian).
3. Baran, P., Burak, К., Kovtun, В., Suhina, А., & Tretiak, К. (2011). Engineering geodetic works in Ukraine. Bulletin of Geodesy and Cartography, (5), 19-26. (in Ukrainian). file:///C:/Users/Admin/Downloads/vgtk_2011_5_6%20(2).pdf
4. Borovyi, V., & Burachek, V. (2017). High-precision engineering-geodetic measurements. Vinnytsia: LLC "Nilan-LTD. (in Ukrainian).
5. Gargula, T. (2009). A special case of the triangle solution with the law of sines in geodetic application. Modern achievements of geodesic science and production, 1(17), 85-91.
6. Goriainov I. (2018) Experimental studies of the use of inverse linear-angular resection to assess the stability of points of a planned deformation geodetic network. Bulletin of SSUGiT, 1, 28-39. (in Ukrainian). https://cyberleninka.ru/article/n/eksperimentalnye-issledovaniya-primene...
7. DBN V.1.3-2:2010. A system for ensuring the accuracy of geometric parameters in construction. Geodetic works in construction. 01.09.2010. Kyiv: Minrehionbud Ukrainy, P. 49. (in Ukrainian). http://online.budstandart.com/ua/ catalog/doc-page?id_doc=25911
8. DSTU-N B V.1.3-1:2009. Performance of measurements, calculation and accuracy control of geometric parameters. Guideline. 01.10.2010. Kyiv: Minrehionbud Ukrainy. P. 71. (in Ukrainian).http://online.budstandart.com/ua/catalog/doc-page?id_doc=25920
9. Moroz, O., Prystupa, O., Shevchenko, T., & Shevchenko, G. (2011). Engineering and geodetic control of the straightness of the axis of the wrapping of the wrapping oven. Geodesy, Cartography, and Aerial Photography, 74, 47-49. (in Ukrainian). https://science.lpnu.ua/istcgcap/all-volumes-and-issues/volume-74-2011/e...
10. Naminat, O. (2020) Improvement of methods of geodesic security for monitoring of linear objects in the zones of inflow of underground gyrnic work. Thesis on the health of the scientific level of the candidate of technical sciences, Lviv, 197. (in Ukrainian).  lpnu.ua/sites/default/ files/2020/ dissertation/ 3806/disnaminatos.pdf
11. Smolii, K. (2015) Analysis of modern geodetic and geotechnical methods for monitoring the deformations of engineering spores. Modern achievements of geodetic science and production. 1. 87-89. (in Ukrainian). http://zgt.com.ua/%d0%b2%d0%b8%d0%bf%d1%83%d1%81%d0%ba-i-29-2015/
12. Staroverov, V., & Gaikin, D. (2020) Geodesic monitoring of hydrotechnical spores for the help of an automated system of guarding. Localization and Territorial Planning: Science-Technology. zb., Kyiv: KNUBA, Issue 74 (in Ukrainian). 298-307. https://doi.org/ 10.32347/2076-815x.2020.74.298-307
13. Trevogo, І. & Balandyuk, А. (2009) Current trends in development and classification of electronic total stations. Modern achievements of geodesic science and production. I (170), 109-115. (in Ukrainian).  vlp.com.ua/ files/20_57.pdf .
14. Trevogo, I., Gorb, А., & Meleshko, О. (2016). Leica MS60 multistation monitoring with high-precision geospatial monitoring. Modern achievements of geodesic science and production, (1), 28-32. (in Ukrainian). http://zgt.com.ua/%d0%b2%d0%b8%d0%bf%d1%83%d1%81%d0%ba-%d1%96-31-2016/
15. Vivat, А. & Nazarchuk, N. (2019). Study of the technique of using the topcon IS 301 scanning total station for the construction of spatial models of architectural forms. Engineering geodesy, (67), 35-45. (in Ukrainian). https://doi.org/10.32347/0130-6014.2019.67.35-45
16. Vivat, A., Tserklevych, A., & Smirnova, O. (2018). A study of devices used for geometric parameter measurement of engineering building construction. Geodesy, Cartography, and Aerial Photography, 87, 21-29. https://doi.org/10.23939/istcgcap2018.01.021