employee
This paper presents an algorithmic complex for geometric modeling and dynamic adaptation of user interfaces in virtual reality (VR) environments, based on methods of engineering geometry and computer graphics. The primary problem addressed is the fundamental inefficiency of traditional 2D design methods, which fail to account for complex spatial relationships and the biomechanical limitations of natural hand movements, often leading to increased cognitive load, physical fatigue, and the risk of cybersickness. The proposed complex is a unified framework that integrates engineering-geometric and computer graphics methods for optimal 3D layout generation and visualization of VR menus, a predictive user behavior model for analysis and forecasting, and reinforcement learning algorithms for dynamic, real-time interface tuning. The core of the methodology is an intelligent system based on Q-learning, which manages interface parameters such as the size, placement, and shape of elements, thus establishing a closed loop of continuous adaptation where each user interaction refines and improves the interface design for future tasks. The feedback for the agent is derived from a multi-objective reward function that carefully balances performance metrics (speed, accuracy) with ergonomic characteristics of user movements, including path efficiency and smoothness (jerk metric). To enhance learning efficiency and guide the exploration of the parameter space, a predictive model based on Gaussian Processes (GPR) is used to forecast user performance and quantify prediction uncertainty, thereby effectively managing the exploration-exploitation trade-off. Experimental validation with 17 students confirmed the approach’s effectiveness, showing a significant reduction in task completion time and high user satisfaction (94.1%). Key metrics, including average task completion time, jerk metric, and path efficiency, showed sustained improvement over the course of the experiment. The results demonstrate that the developed complex provides a robust framework for creating ergonomic and efficient VR interfaces, which is particularly valuable for educational and training applications where long-term, comfortable use is critical for successful learning and minimizing health risks.
geometric modeling, multidimensional space, Simplex, Apollonius problem, logic programming language, Prolog, predicate
1. Boikov A.A., Kadykova N.S. Inzhenernaya geometriya kak fundamental’noye yadro inzhenernoy podgotovki spetsialistov [Engineering Geometry as a Fundamental Core of Engineering Training of Specialists]. Geometriya i grafika [Geometry and graphics]. 2024, V. 12, I. 4, pp. 15–37. DOI:https://doi.org/10.12737/2308-4898-2024-12-4-15-37. (in Russian)
2. Kovalev A.I., Rogoleva Yu.A., Egorov S.Yu. Sravnenie effektivnosti primeneniya tekhnologiy virtual’noy real’nosti s traditsionnymi obrazovatel’nymi sredstvami [Comparison of the Effectiveness of Virtual Reality Technologies with Traditional Educational Tools]. Vestnik Moskovskogo universiteta. Seriya 14. Psikhologiya [Bulletin of Moscow University. Series 14. Psychology]. 2019, No. 4, pp. 44–58. DOI:https://doi.org/10.11621/vsp.2019.04.43. (in Russian)
3. Krupoderova E.P., Krupodyorova K.R. Podgotovka budushchikh magistrov pedagogicheskogo obrazovaniya k ispol’zovaniyu immersivnykh tekhnologiy obucheniya [Preparing Future Master’s Students of Pedagogical Education to Use Immersive Learning Technologies]. Problemy sovremennogo pedagogicheskogo obrazovaniya [Problems of Modern Pedagogical Education]. 2022, I. 77-3, pp. 105–108. (in Russian)
4. Leparov M.N. O geometricheskikh osnovakh proektirovaniya tekhnicheskogo obyekta [On the Geometric Foundations of Designing a Technical Object]. Geometriya i grafika [Geometry and graphics]. 2023, V. 11, I. 4, pp. 3–14. DOI:https://doi.org/10.12737/2308-4898-2024-11-4-3-14. (in Russian)
5. Lubin Sh. Ispol’zovanie tsifrovykh prilozheniy v protsesse izucheniya inostrannykh yazykov [Use of Digital Applications in the Process of Studying Foreign Languages]. Yaroslavskiy pedagogicheskiy vestnik [Yaroslavl Pedagogical Bulletin]. 2024, I. 3, pp. 100–106. DOI:https://doi.org/10.20323/1813145X-2024-3-138-100. (in Russian)
6. Muray O.V. Obuchenie inostrannym yazykam s pomoshch’yu VR-tekhnologii [Teaching Foreign Languages Using VR Technology]. Mir nauki, kul’tury, obrazovaniya [World of Science, Culture, Education]. 2023, I. 5, pp. 189–192. DOI:https://doi.org/10.24412/1991-5497-2023-5102-189-192. (in Russian)
7. Nesnov D.V. Oblast’ pravil’noy koordinatsii prostranstva normal’nymi konicheskimi koordinatami [Region of Proper Coordination of Space by Normal Conic Coordinates]. Geometriya i grafika [Geometry and graphics]. 2023, V. 11, I. 3, pp. 3–11. DOI:https://doi.org/10.12737/2308-4898-2023-11-3-3-11. (in Russian)
8. Rustamyan V.V., Bayanov E.V., Slavin R.B. Sinteticheskoye predstavlenie preobrazovaniya ‘kosaya simmetriya’ na primere preobrazovaniya ellipsa [Synthetic Representation of the Transformation ‘Oblique Symmetry’ on the Example of Transforming an Ellipse]. Geometriya i grafika [Geometry and graphics]. 2023, V. 11, I. 3, pp. 12–18. DOI:https://doi.org/10.12737/2308-4898-2023-11-3-12-18. (in Russian)
9. Salkov N.A. Opredelenie rasstoyaniy mezhdu geometricheskimi figurami interaktivnym metodom [Determination of Distances Between Geometric Figures by an Interactive Method]. Geometriya i grafika [Geometry and graphics]. 2024, V. 12, I. 4, pp. 3–14. DOI:https://doi.org/10.12737/2308-48982024-12-4-3-14. (in Russian)
10. Sidorov K.Yu., Ivanova E.A. Vizualizatsiya i vzaimodeystvie s geometricheskimi ob’ektami v VR-srede [Visualization and Interaction with Geometric Objects in a VR Environment]. Geometriya i grafika [Geometry and graphics]. 2024, V. 12, I. 3, pp. 45–53. DOI:https://doi.org/10.12737/2308-4898-2024-123-45-53. (in Russian)
11. Chupin S.A., Isaev A.I., Trubetskaya O.V. Primenenie dopolnennoy real’nosti pri prepodavanii inzhenernoy gorno-geologicheskoy grafiki [Use of Augmented Reality in Teaching Engineering Mining-Geological Graphics]. Geometriya i grafika [Geometry and graphics]. 2024, V. 12, I. 3, pp. 21–31. DOI:https://doi.org/10.12737/2308-4898-2024-12-3-21-31. (in Russian)
12. Khusainova M.A., Garifova O.A. Integratsiya tekhnologiy virtual’noy real’nosti v protsess obucheniya inostrannym yazykam: effektivnost’ i metodicheskiye podkhody [Integration of Virtual Reality Technologies into Foreign Language Teaching: Effectiveness and Methodological Approaches]. Mezhdunarodnyy nauchno-issledovatel’skiy zhurnal [International Research Journal]. 2024, I. 11, pp. 92–97. DOI:https://doi.org/10.60797/IRJ.2024.149.95. (in Russian)
13. Alcañiz M., Bigné E., Guixeres J. Virtual reality in marketing: a framework, review, and research agenda Frontiers in Psychology. 2019. V. 10, p. 1530. DOI:https://doi.org/10.3389/fpsyg.2019.01530
14. Ashrafi D.M. Managing consumers’ adoption of artificial intelligence-based financial robo-advisory services: a moderated mediation model. Journal of Indonesian Economy & Business. 2023, V. 38, I. 3, pp. 204–220. DOI: 10.22146/ jieb.v38i3.6242
15. Bailenson J.N. Experience on Demand: What Virtual Reality Is, How It Works, and What It Can Do. W.W. Norton & Company, 2018, 320 p.
16. Berthaut F. 3D interaction techniques for musical expression. Journal of New Music Research. 2020, V. 49, I. 1, pp. 60–72. DOI:https://doi.org/10.1080/09298215.2019.1706584
17. Berwal P., Dhatterwal J.S., Kaswan K.S., Kant S. Computer Applications in Engineering and Management. Chapman and Hall/CRC, 2022, 250 p. ISBN: 9781032228716.
18. Bowman D.A., Kruijff E., LaViola J.J., Poupyrev I. An introduction to 3-D user interface design. Presence. 2001, pp. 96–108. DOI:https://doi.org/10.1162/105474601750182342
19. Browning T.R. Design structure matrix extensions and innovations: a survey and new opportunities. IEEE Transactions on Engineering Management. 2015, V. 63, I. 1, pp. 27–52. DOI:https://doi.org/10.1109/TEM.2015.2491283
20. Fitts P.M. The information capacity of the human motor system in controlling the amplitude of movement. Journal of Experimental Psychology. 1954, V. 47, I. 6, pp. 381–391. DOI:https://doi.org/10.1037/h0055392
21. Hincapie C., Lécuyer A. A survey on adaptive virtual reality interfaces. IEEE Transactions on Visualization and Computer Graphics. 2018, V. 24, I. 1, pp. 295–306. DOI:https://doi.org/10.3390/s24196204
22. Horváth I. An analysis of personalized learning opportunities in 3D VR. Frontiers in Computer Science. 2021, V. 3, p. 673826. DOI:https://doi.org/10.3389/fcomp.2021.673826
23. LaViola J.J. Jr., Kruijff E., McMahan R.P., Bowman D., Poupyrev I.P. 3D User Interfaces: Theory and Practice. Addison-Wesley Professional, 2017, 600 p.
24. Lee Y., Yoo B. XR collaboration beyond virtual reality: work in the real world. Journal of Computational Design and Engineering. 2021, V. 8, I. 2, pp. 756–772. DOI:https://doi.org/10.1093/jcde/qwab012
25. MacKenzie I.S. Fitts’ law as a research and design tool in human–computer interaction. Human-Computer Interaction. 1992, V. 7, I. 1, pp. 91–139. DOI: 10.1207/ s15327051hci0701_3.
26. Nturner J., Weyers B. Towards formalisation of 3D user interaction: making spatial interaction verifiable. In Companion Proceedings of the 17th ACM SIGCHI Symposium on Engineering Interactive Computing Systems. 2025, pp. 27–32. DOI:https://doi.org/10.1145/3731406.3734976.
27. Okpan J.I., Brooks R.J. Experimental investigation of the impacts of virtual reality on discrete-event simulation. Proceedings – Winter Simulation Conference. 2005, pp. 19671975. DOI:https://doi.org/10.1109/WSC.2005.1574475
28. Pind R., Böhmer M., Rädle R., Hincapié C., Weinhardt C. A framework for the design and evaluation of VR applications. Frontiers in Virtual Reality. 2021, V. 2, p. 698123. DOI:https://doi.org/10.1145/502390.502405
29. Poupyrev I., Billinghurst M., Weghorst S., Ichikawa T. The Go-Go interaction technique: non-linear mapping for direct manipulation in VR. Proceedings of the 9 th Annual ACM Symposium on User Interface Software and Technology (UIST ’96). 1996, pp. 79–80. DOI:https://doi.org/10.1145/237091.237102
30. Suter S., Fröhlich B. Design and evaluation of a dynamic menu system for immersive virtual environments. Journal of Virtual Reality and Broadcasting. 2010, V. 7, I. 3, pp. 1–15. DOI:https://doi.org/10.1145/3489849.3489890.
31. Teather R.J., Stuerzlinger W. Pointing at 3D target projections with one-eyed and stereo cursors. Proceedings of thehttps://doi.org/10.1145/3733155.3736593
32. Wang Z., Rao M., Ye S., Song W., Lu F. Towards spatial computing: recent advances in multimodal natural interaction for XR headsets. arXiv preprint. 2025, arXiv:2502.07598
33. Yeo A., Kwok B.W., Joshna A., Chen K., Lee J.S. Entering the next dimension: a review of 3D user interfaces for virtual reality. Electronics. 2024, V. 13, I. 3, p. 600. DOI:https://doi.org/10.3390/electronics13030600
34. Zhai S., Kristensson P.O. The word-gesture keyboard: reimagining keyboard interaction. Communications of the ACM. 2012, V. 55, I. 9, pp. 91–101. DOI:SIGCHI Conference on Human Factors in Computing Systems. 2013, pp. 159–168. DOI:https://doi.org/10.1145/2470654.247067
35. Zouhaier L., Hlaoui Y.B.D., Ayed L.B. A reinforcement learning based approach of context-driven adaptive user interfaces. In 2021 IEEE 45th Annual Computers, Software, and Applications Conference (COMPSAC). 2021, pp. 1463–1468. DOI:https://doi.org/10.1109/COMPSAC51774.2021.00217
