Multidisciplinary Journal of Science and Technology

GEOMETRIC MODELING AND ANALYSIS OF COMPLEX-SHAPED OBJECTS USING THE R-FUNCTION METHOD

Nuraliyev Fakhriddin

Professor of the Department of "Television and Media Technologies" of the Tashkent University of Information Technologies named after Muhammad al-Khwarizmi

Gulmirzayeva Zulayho

Tashkent University of Information Technologies named after Muhammad al-Khwarizmi,

Keywords: R-functions, geometric modeling, complex shapes, 3D objects, functional representation, constructive solid geometry, virtual environment.

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Abstract

This article discusses the issues of geometric modeling of objects with complex shapes using the R-functions method. The theory of R-functions, developed by V.L. Rvachev, allows representing logical operations (union, intersection, difference) through continuous real functions. This method enables describing complex three-dimensional objects with a single analytical function based on simple primitives. As a result, the object's boundary and interior become smooth (with C¹ or higher continuity), which facilitates application in virtual environments, educational platforms, and computer graphics systems. The article analyzes the main properties of R-functions, the process of constructive logical-algebraic modeling, and methods for creating complex shapes (such as interior and exterior design elements, virtual objects). The proposed approach aims to increase the efficiency of 3D model creation in virtual educational environments.

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References

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2.Rvachev, V. L. (1967). Geometricheskie prilozheniya bulevoy algebry. Kiev: Tekhnika. (rus tilida)

3.Rvachev, V. L. (1982). Teoriya R-funktsiy i nekotorye ee prilozheniya. Kiev: Naukova Dumka. (rus tilida)

4.Shapiro, V. (2007). Semi-analytic geometry with R-functions. Acta Numerica, 16, 239–303. https://doi.org/10.1017/S096249290629001X

5. Pasko, A., Adzhiev, V., Sourin, A., & Savchenko, V. (1995). Function representation in geometric modeling: Concepts, implementation and applications. The Visual Computer, 11(8), 429–446. https://doi.org/10.1007/BF02464333

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7.Tolok, A., & Tolok, N. (2021). Local R-Functional Modelling (LRFM). In GraphiCon 2021: 31st International Conference on Computer Graphics and Vision (pp. 1–10). CEUR Workshop Proceedings.

8.Kucherenko, S., Shah, N., & Klymenko, O. (2025). Identification of feasible regions using R-functions. arXiv preprint arXiv:2503.05510.

9.Lisnyak, A. A. (2009). Application of R-functions for geometric modeling of complex-shaped objects. Radio Electronics, Computer Science, Control, (1), 112–118.

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Ushbu ro‘yxat maqola mavzusiga (R-funksiyalar usuli yordamida murakkab shakllarni geometrik modellashtirish) mos ravishda tuzilgan bo‘lib, asosiy nazariy asarlar (Rvachev), zamonaviy sharhlar (Shapiro, Pasko) va amaliy qo‘llanishlar bilan bog‘liq ishlar kiritilgan.

Kitoblar

1. Rvachev, V. L. (1967). Geometricheskie prilozheniya bulevoy algebry. Kiev: Tekhnika.

2. Rvachev, V. L. (1982). Teoriya R-funktsiy i nekotorye ee prilozheniya. Kiev: Naukova Dumka.

Jurnallar

1. Pasko, A., Adzhiev, V., Sourin, A., & Savchenko, V. (1995). Function representation in geometric modeling: Concepts, implementation and applications. The Visual Computer, 11(8), 429–446. https://doi.org/10.1007/BF02464333

2. Shapiro, V. (2007). Semi-analytic geometry with R-functions. Acta Numerica, 16, 239–303. https://doi.org/10.1017/S096249290629001X

3. Lisnyak, A. A. (2009). Application of R-functions for geometric modeling of complex-shaped objects. Radio Electronics, Computer Science, Control, (1), 112–118. https://doi.org/10.15588/1607-3274-2009-2-14

4. Kucherenko, S., Shah, N., & Klymenko, O. (2025). Identification of feasible regions using R-functions. arXiv preprint arXiv:2503.05510.

5. Tolok, A., & Tolok, N. (2021). Local R-Functional Modelling (LRFM). In GraphiCon 2021: 31st International Conference on Computer Graphics and Vision. CEUR Workshop Proceedings.

Veb-sayt

1.HyperFun Project. (n.d.). Function Representation (FRep) modeling. Retrieved from http://www.hyperfun.org