Mathematical model of the formation of cyclic and channel surfaces based on nonlinear rotation
https://doi.org/10.25206/1813-8225-2024-192-14-21
EDN: PHLZOO
Abstract
This work is a continuation of the series of works by the authors devoted to the issues of shaping surfaces of nonlinear rotation. The geometric scheme for the formation of surfaces of this class includes: an axis of nonlinear rotation, which is a smooth, generally spatial curve, and a forming line, also a smooth spatial curve. When the generating line rotates relative to the curvilinear axis, each point of the generating line describes a circumferential trajectory in the corresponding normal plane of the rotation axis. As a result, a surface of nonlinear rotation is formed, which is a normal cyclic surface. In this work, in order to develop the research results previously obtained by the authors in the field of shaping surfaces of nonlinear rotation, the solution to the inverse problem of shaping is considered and a mathematical justification is given for the possibility of shaping a channel surface based on solutions to the direct and inverse problems. The work provides numerical examples of the formation of the surfaces under consideration, accompanied by mathematical models of surfaces, their computer implementation and visualization. The research results can be useful in the development of CAD systems that involve the design of surface forms of products based on cyclic and channel surfaces in mechanical engineering, construction, architecture and other practical fields.
About the Authors
K. L. Panchuk
Omsk State Technical University
Russian Federation
Russian Federation
PANCHUK Konstantin Leonidovich, Doctor of Technical Sciences, Associate Professor, Professor of Engineering Geometry and CAD Department
Omsk
AuthorID (RSCI): 501163
AuthorID (SCOPUS): 55857766100
ResearcherID: S-2788-2017
T. M. Myasoedova
Omsk State Technical University
Russian Federation
Russian Federation
MYASOEDOVA Tatyana Mikhaylovna, Candidate of Technical Sciences, Senior Lecturer of Engineering Geometry and CAD Department
Omsk
AuthorID (RSCI): 686836
AuthorID (SCOPUS): 57201776004
ResearcherID: E-7505-2014
E. V. Lyubchinov
Omsk State Technical University
Russian Federation
Russian Federation
LYUBCHINOV Evgeniy Vladimirovich, Candidate of Technical Sciences, Associate Professor of Engineering Geometry and CAD Department
Omsk
AuthorID (RSCI): 917932
AuthorID (SCOPUS): 57199399265
ResearcherID: D-1882-2019
References
1. Panchuk K. L., Myasoyedova T. M., Lyubchinov E. V. Tsiklicheskiye poverkhnosti, soprovozhdayushchiye nelineychatyye kvadriki vrashcheniya [Cyclic surfaces accompanying non-ruled quadrics of rotation] // Omskiy nauchnyy vestnik. Omsk Scientific Bulletin. 2023. No. 3 (187). P. 23–29. DOI: 10.25206/1813-8225-2023-187-23-29. EDN: BAKBPA. (In Russ.).
2. Panchuk K. L., Myasoyedova T. M. Poverkhnost’ nelineynogo vrashcheniya [The surface of non-linear rotation] // Omskiy nauchnyy vestnik. Omsk Scientific Bulletin. 2023. No. 4 (188). P. 5–12. DOI: 10.25206/1813-8225-2023-188-5-12. EDN: VFCPSL. (In Russ.).
3. Grigor’yev M. I. Postroyeniye obobshchennykh poverkhnostey vrashcheniya [Construction of generalised rotation surfaces] // Seminar «DNA&CAGD». Izbrannyye doklady. Seminar «DNA&CAGD». Izbrannyye Doklady. 2007. P. 1–7. (In Russ.).
4. Grigor’yev M. I., Malozemov V. N. Sostavnyye krivyye i poverkhnosti Bez’ye. Analiticheskiy podkhod [Compound curves and Bézier surfaces. Analytical approach]. Lambert Academic Publishing, 2010. 132 p. ISBN 978-3-8433-0323-1. (In Russ.).
5. Beglov I. A., Rustamyan V. V. Metod vrashcheniya geometricheskikh ob”yektov vokrug krivolineynoy osi [Method of rotation of geometrical objects around the curvilinear axis] // Geometriya i grafika. Geometry and Graphics. 2017. No. 3. P. 45– 50. DOI: 10.12737/article_59bfa4eb0bf488.99866490. (In Russ.).
6. Beglov I. A. Computer geometric modeling of quasirotation surfaces // Journal of Physics: Conference Series. 2021. Vol. 1901. P. 16–17. DOI: 10.1088/1742-6596/1901/1/012057. (In Engl.).
7. Peternell M., Pottmann H. Computing Rational Parametrizations of Canal Surfaces // Journal Symbolic Comput. 1997. Vol. 23, Issue 2–3. P. 255–266. DOI: 10.1006/jsco.1996.0087. (In Engl.).
8. Xu Z., Feng R., Sun J. G. Analytic and Algebraic Properties of Canal Surfaces // Journal of Computational and Applied Mathematics. 2006. Vol. 195, Issues 1–2. P. 220–228. DOI: 10.1016/j.cam.2005.08.002. (In Engl.).
9. Öztürk G., Bulca B., Bayram B. K. [et al.]. On Canal Surfaces in E3 // Selçuk Journal of Applied Mathematics. 2010. Vol. 11, no. 2. P. 103–108. (In Engl.).
10. Kim Y. H., Liu H., Qian J. Some characterizations of canal surfaces // Bull. Korean Math. Soc. 2016. Vol. 53 (2). P. 461–477. DOI: 10.4134/BKMS.2016.53.2.461. (In Engl.).
11. Krivoshapko S. N., Ivanov V. N. Entsiklopediya analiticheskikh poverkhnostey (boleye 500 poverkhnostey, 38 klassov: matematikam, inzheneram, arkhitektoram) [Encyclopaedia of Analytical Surfaces (over 500 surfaces, 38 classes: for mathematicians, engineers, architects)]. Moscow, 2010. 556 p. ISBN 978-5-397-00985-0. (In Russ.).
12. Farouki R. A., Sverrissor R. Approximation of Rollingball Blends for Freeform Parametric Surfaces // Computer-Aided Design. 1996. Vol. 28 (11). P. 871–878. EDN: AKQVYD. (In Engl.).
13. Hartman E. Geometry and Algorithms for Computer Aided Design. Department of Mathematics Darmstadt University of Technology. Darmstadt, Germany, 2003. 160 p. (In Engl.).
14. Nyi N. H., Kyaw H., Markin L. V. Issledovaniye algoritmov ispol’zovaniya retseptornykh geometricheskikh modeley v zadachakh telesnoy trassirovki aviatsionnoy tekhniki [The study of algorithms using receptor geometric model in the problems of physical route tracing in aviation technology] // Trudy MAI. Trudy MAI. 2013. No. 69. P. 1–25. URL: www.mai.ru/upload/iblock/62a/62ad38934954abb6876b2d621d39098f.pdf (accessed: 12.03.2024). (In Russ.).
15. Markin L. V. Geometricheskiye modeli avtomatizirovannoy komponovki letatel’nykh apparatov [Geometric Models of Automated Layout Aircrafts] // Vestnik MAI. Aerospace MAI Jornal. 2015. Vol. 22, no. 1. P. 47–56. EDN: TNWXMF. (In Russ.).
16. Nyi N. H., Markin L. V., Sosedko A. A. Primeneniye retseptornykh geometricheskikh modeley v zadachakh avtomatizirovannoy komponovki aviatsionnoy tekhniki [Implementation of receptor geometric models in the problems of automated layout design in aviation technology] // Trudy MAI. Trudy MAI. 2014. No. 72. P. 1–26. URL: http://www.mai.ru/upload/iblock/f17/f178d8a078b6927a66bd134f8a37e7ad.pdf (accessed: 12.03.2024). (In Russ.).
17. Ma Y., Tu C., Wang W. Computing the Distance between Canal Surfaces // Advances in Geometric Modeling and Processing. GMP 2010. Lecture Notes in Computer Science. 2010. Vol. 6130 / Eds. B. Mourrain, S. Schaefer, G. Xu. Springer, Berlin, Heidelberg. DOI: 10.1007/978-3-642-13411-1_7. (In Engl.).
18. Núñez-Valdés J., Ocaña Almagro I. Canal surfaces and its application to the CAGD // American Journal of Engineering Research (AJER). 2021. Vol. 10, Issue 03. P. 19–31. (In Engl.).
Review
For citations:
Panchuk K.L., Myasoedova T.M., Lyubchinov E.V. Mathematical model of the formation of cyclic and channel surfaces based on nonlinear rotation. Omsk Scientific Bulletin. 2024;(4):14-21. (In Russ.) https://doi.org/10.25206/1813-8225-2024-192-14-21. EDN: PHLZOO
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