Research of the differential properties of the navigational isosurface

The topic of this paper is a qualitative study of navigational isosurfaces to establish the practicality of polynomial approximation for a class of differentiable functions with minimized “smoothness.” When solving problems of restoring the scalar field of navigation parameters, it is certainly important to form an initial correct judgment about the isogeometric proximity between the approximate and approximating functions, provided specific information is given about the differential properties of the synthesized navigation isosurface. It is assumed that the structure of the graphical model of the object under study and the characteristics of function approximation theory should be consistent with each other when forming a unified information approach. As a concrete illustrative example, a study of the differential properties of a formalized representation of an astronavigational isosurface — with geometric interpretation in the form of computer screenshots obtained as a result of the work of a compiled software module — has been performed. An assumption is made regarding the realistic possibility of choosing the optimal approximate navigation function based on visualization of the “smoothness” of a navigation function of any dimension in accordance with Schoenberg’s hypothesis about the relationship between minimum curvature and maximum smoothness of an algebraic line. The search for a solution to the problem of graphical transformation of break points of an abstract isoline, formalized in strict mathematical terms as a special case of a navigational isosurface, is determined. The developed methodology is proposed for effective verification of the reliability of big geospatial data. The prospective importance of proper mathematical processing of marine spatial data from geographic information systems is emphasized as a priority in providing consumers with reliable information for practical purposes. The relevance of the need to conduct a qualitative study of navigational isosurfaces to successfully approximate them from the unified standpoint of spline function theory coincides with the prediction of the emergence of complex hypothetical isolines, which are expected to be studied during the evolution of technical means of navigation.

1. Kvasov, B. I. Metody izogeometricheskoy approksimatsii splaynami. Abstract of Grand PhD diss. Novosibirsk, 1997.

2. Kvasov, B. I. Methods of Shape-Preserving Spline Approximation Singapore: World Scientific Publishing Co. Pte. Ltd., 2000. DOI: 10.1142/4172.

3. Yuyukin, I. V. “Modification of the least squares method for spline approximation of navigational isosurface.” Vestnik gosudarstvennogo universiteta morskogo i rechnogo flota im. admirala S. O. Makarova 11.4 (2019): 631–639. DOI: 10.21821/2309-5180-2019-11-4-631-639.

4. Boltaev, A. K. “On the optimal interpolation formula on classes of differentiable functions.” Problemy vychislitel’noy i prikladnoy matematiki 4(34) (2021): 96–105.

5. Lapshin, E. V., I. Yu. Semochkina and V. V. Samarov. “Kusochno-lineynaya interpolyatsiya funktsiy mnogikh argumentov.” Reliability & Quality Of Complex Systems 4(20) (2017): 42–48. DOI: 10.21685/2307-42052017-4-6.

6. Yuyukin, I. V. “Application of spline interpolating functions in the paradigm of the universal standard exchange of digital hydrographic data.” Vestnik gosudarstvennogo universiteta morskogo i rechnogo flota im. admirala S. O. Makarova 14.6 (2022): 875–890. DOI: 10.21821/2309-5180-2022-14-6-875-890.

7. Starodetko, E. A. Metody opisaniya i preobrazovaniya geometrisheskoy informatsii v avtomatizirovannyh sistemah tehnisheskoy podgotovki proizvodstva. Abstract of Grand PhD diss. M., 1974.

8. Vasil’eva, E. V. “Stability of periodic points of diffeomorphisms of multidimensional space.” Vestnik Of Saint Petersburg University. Mathematics. Mechanics. Astronomy 5.3 (2018): 356–366. DOI: 10.21638/11701/Spbu01.2018.302.

9. Kurakhtenkov, L. V., A. A. Kuchumov and K. Yu. Shkil’. “Razrabotka algoritma postroeniya izoliniy po rasschitannym dannym na sfericheskoy poverkhnosti.” T-Comm 11.5 (2017): 21–25.

10. Wang, W., L. Zhou, A-X. Zhu and G. Lv. “Isoline extraction based on a global hexagonal grid.” Cartography and Geographic Information Science 52.3 (2025): 299–313. DOI: 10.1080/15230406.2024.2359709.

11. Smith, W. H. F., and P. Wessel. “Gridding with Continuous Curvature Splines in Tension.” Geophysics 55.3 (1990): 293–305. DOI: 10.1190/1.1442837.

12. Yuyukin, I. V. “Spline model of gridded data operation as a principle of electronic mapping seabed topography.” Vestnik gosudarstvennogo universiteta morskogo i rechnogo flota im. admirala S. O. Makarova 14.5 (2022): 656–675. DOI: 10.21821/2309-5180-2022-14-5-656-675.

13. Costan, Alexandru. “Grid Data Handling.” Computational and Data Grids: Principles, Applications and Design. Hershey: Information Science Reference, 2012. 112–139. DOI: 10.4018/978-1-61350-113-9.ch005.

14. Souri, Amir H., Kelly Chance, Kong Sun, Xiong Liu, and Matthew S. Johnson. “Dealing with spatial heterogeneity in pointwise-to-griddeddata comparisons.” Atmospheric Measurement Techniques 15.1 (2022): 41–59. DOI: 10.5194/amt-15-41-2022.

15. Yuyukin, I. V. “Cubic splines synthesis of a distorted isoline in the aspect of using differential mode of satellite navigation.” Vestnik gosudarstvennogo universiteta morskogo i rechnogo flota im. admirala S. O. Makarova 13.3 (2021): 341–358. DOI: 10.21821/2309-5180-2021-13-3-341-358.

16. Kvasov, B. and T-W. Kim. “Weighted cubic and biharmonic splines.” Computational Mathematics and Mathematical Physics 57.1 (2017): 26–44.

17. Taheri, A. H. and K. Suresh. “Adaptive w-refinement: A new paradigm in isogeometric analysis.” Computer Methods in Applied Mechanics and Engineering 368 (2020): 113180. DOI: 10.1016/j.cma.2020.113180.

18. Schoenberg, I. J. “Contribution to the Problem of Approximation of Equidistant Data by Analytic Functions.” Quarterly of Applied Mathematics 4.1 (1946): 45–99. DOI: 10.1090/qam/15914.

19. Yuyukin, I. V. “Generalization of the underwater relief image using the spline approximation method on a vector electronic chart.” Vestnik gosudarstvennogo universiteta morskogo i rechnogo flota im. admirala S. O. Makarova 16.6 (2024): 910–934. DOI: 10.21821/2309-5180-2024-16-6-910-934.

20. Yuyukin, I. V. “Spline interpolation of navigational isolines.” Vestnik gosudarstvennogo universiteta morskogo i rechnogo flota im. admirala S. O. Makarova 11.6 (2019): 1026–1036. DOI: 10.21821/2309-5180-2019-116-1026-1036.

21. Volkov, Yu. S. and V. V. Bogdanov. “On error estimates of local approximation by splines.” Sibirskii Matematicheskii Zhurnal 61.5 (2020): 1001–1008. DOI: 10.33048/smzh.2020.61.503.

22. Wang, H., C-G. Zhu and C-Y. Li. “Construction of B-spline surface from cubic B-spline asymptotic quadrilateral.” Journal of Advanced Mechanical Design, Systems, and Manufacturing 11.4 (2017): JAMDSM0044. DOI: 10.1299/jamdsm.2017jamdsm0044.

23. Vorob’eva, G. R. and E. F. Farvaev. “On the issue of constructing spatial isolines for irregular monitoring networks.” Information Technologies 30.10 (2024): 544–552. DOI: 10.17587/it.30.544-552.

24. Yuyukin, I. V. “Search for errors in the base of navigation data by the method of spline isosurface visualization.” Vestnik gosudarstvennogo universiteta morskogo i rechnogo flota im. admirala S. O. Makarova 12.3 (2020): 481–491. DOI: 10.21821/2309-5180-2020-12-3-481-491.

25. Morrison J. L. “Cartography of the new millennium.” Geodesy and cartography 12.3 (1996): 45–48.

26. Loukili, Y., Y. Lakhrissi and S. E. B. Ali. “Geospatial big data platforms: A comprehensive review.” KN — Journal of Cartography and Geographic Information 72.4 (2022): 293–308. DOI: 10.1007/s42489-022-00121-7.

27. Andrienko, N. and G. Andrienko. “Geovisualization: past, present, and future.” International Journal of Cartography 11.2 (2025): 224–234. DOI: 10.1080/23729333.2025.2475410.

28. Li, Z. “Geospatial Big Data Handling with High Performance Computing: Current Approaches and Future Directions.” High Performance Computing for Geospatial ApplicationsSpringer International Publishing, 2020: 53–76. DOI: 10.1007/978-3-030-47998-5_4.

29. Robinson, A. C., C. R. Sluter, et al. “Geospatial big data and cartography: research challenges and opportunities for making maps that matter.” International Journal of Cartography 3.sup1 (2017): 32–60. DOI: 10.1080/23729333.2016.1278151.

30. Jamil, D., M. A. Mondal, et al. “Exploring Geospatial Mapping through Speech Commands.” 2025 International Conference on Computer, Electrical & Communication Engineering (ICCECE)2025: 1–7. DOI: 10.1109/ICCECE61355.2025.10940999.