Moskva, Moscow, Russian Federation
. This article presents the results of a study of the geometric properties of the Nicomed conchoid and the oblique conchoid. In this paper, the oblique conchoid is modeled in a new way, namely by quasi-symmetry with respect to the elliptic axis. The method used is a fourth-order transformation of the plane relative to the second-order curve. That is, a straight line with quasi-symmetry is mapped into a fourth-order curve. The image of a straight line in this case consists of two branches that tend to two asymptotes. Quasi–symmetry makes it possible to obtain an oblique conchoid, as a special case under certain conditions, and in the general case, many other conchoidal curves. The use of this method made it possible to discover new geometric properties of conchoidal curves, in particular, to find a previously undescribed constructive correspondence between points belonging to different branches of the oblique conchoid. The paper formulates and proves three statements, namely: 1) The image of a straight line with its quasi-symmetry with respect to a circle is a Nicomedes conchoid, 2) the image of a circle with its quasi-symmetry with respect to a circle is a curve of the sixth order, 3) the image of a straight parallel major semiaxis of an ellipse with its quasi-symmetry with respect to a given ellipse is two symmetrical oblique conchoids with respect to the minor semiaxis of an ellipse. Also, the equations of the curves under consideration and their asymptotes in the general case are derived. The results of the research carried out in this paper expand the possibilities of using conchoidal curves in solving problems of engineering geometry. For example, when modeling various physical phenomena and processes, as well as in engineering and architectural design.
: quasi-symmetry, quasi-rotation, Nicomed's conchoid, oblique conchoid, conchoidal curves
1. Antonova I.V., Beglov I.A., Solomonova E.V. Matematicheskoe opisanie vrashheniya tochki vokrug e`llipticheskoj osi v nekotory`x chastny`x sluchayax [Mathematical description of the rotation of a point around an elliptical axis in some particular cases]. Geometriya i grafika [Geometry and Graphics]. 2019, V. 7, I. 3. pp. 36-50. DOI:https://doi.org/10.12737/article_5dce66dd9fb966.59423840. (in Russian)
2. Beglov I.A. Atlas poverhnostej kvazivrashcheniya [Atlas of quasi-rotation surfaces]. Moscow, Infra-M Publ., 2022. 76 p. (in Russian)
3. Beglov I.A., Rustamyan V.V., Antonova I.V. Matematicheskoe opisanie metoda vrashcheniya tochki vokrug krivolinejnoj osi vtorogo poryadka [Mathematical description of the method of rotation of a point around a curved axis of the second order]. Geometriya i grafika [Geometry and Graphics]. 2018, V. 6, I. 3. pp. 39-46. DOI:https://doi.org/10.12737/article_5c21f6e832b4d2.25216268. (in Russian)
4. Beglov I.A., Rustamyan V.V. Metod vrashcheniya geometricheskih ob"ektov vokrug krivolinejnoj osi [Method of rotation of geometric objects around a curved axis]. Geometriya i grafika [Geometry and Graphics]. 2017, V. 5, I. 3, pp. 45-50. DOI:https://doi.org/10.12737/article_59bfa4eb0bf488.99866490. (in Russian)
5. Bojkov A.A. Razrabotka i primenenie yazyka geometricheskih postroenij dlya sozdaniya komp'yuternyh geometricheskih modelej [Development and application of the language of geometric constructions for the creation of computer geometric models]. Problemy mashinovedeniya: materialy V Mezhdunarodnoj nauchno-tekhnicheskoj konferencii [Problems of machine science: materials of the V International Scientific and Technical Conference]. 2021, pp. 423-429. DOIhttps://doi.org/10.25206/978-5-8149-3246-4-2021-423-429. (in Russian)
6. Bermant, A.F. Geometricheskij spravochnik po matematike. Atlas krivyh [Geometric Handbook of Mathematics. Atlas of Curves]. Moscow, ONGIZ NKTP V Publ., 1937. 209 p. (in Russian)
7. Vyshnepol'skij V.I., Zavarihina E.V., Pekh D.S. Geometricheskie mesta tochek, ravnootstoyashchih ot dvuh zadannyh geometricheskih figur. chast' 4: geometricheskie mesta tochek, ravnoudalennyh ot dvuh sfer [Geometric locations of points equidistant from two given geometric shapes. part 4: geometric locations of points equidistant from two spheres]. Geometriya i grafika [Geometry and Graphics]. 2021, V. 9, I. 3, pp. 12-29. DOI:https://doi.org/10.12737/2308-4898-2021-9-3-12-29. (in Russian)
8. Vyshnepol'skij V.I., Zavarihina E.V., Egiazaryan K.T. Geometricheskie mesta tochek, ravnootstoyashchih ot dvuh zadannyh geometricheskih figur. chast' 5: geometricheskie mesta tochek, ravnoudalennyh ot sfery i ploskosti [Geometric locations of points equidistant from two given geometric shapes. part 5: geometric locations of points equidistant from the sphere and plane]. Geometriya i grafika [Geometry and Graphics]. 2021, V. 9, I. 4, pp. 22-34. DOI:https://doi.org/10.12737/2308-4898-2022-9-4-22-34. (in Russian)
9. Ivanov G.S. Teoreticheskie osnovyi nachertatelnoy geometrii [Theoretical Foundations of Descriptive Geometry]. Moscow, Mechanical engineering Publ., 1998. 157 p. (in Russian)
10. Korotkij V.A., Vitovtov I.G. Approksimaciya fizicheskogo splajna s bol'shimi progibami [Approximation of a physical spline with large deflections]. Geometriya i grafika [Geometry and Graphics]. 2021, V. 9, I. 1, pp. 3-19. DOI:https://doi.org/10.12737/2308-4898-2021-9-1-3-19. (in Russian)
11. Korotkij V.A. Konstruirovanie G2-gladkoj sostavnoj krivoj na osnove kubicheskih segmentov Bez'e [Construction of a G2-smooth composite curve based on cubic Bezier segments]. Geometriya i grafika [Geometry and Graphics]. 2021, V. 9, I. 2, pp. 12-28. DOI:https://doi.org/10.12737/2308-4898-2021-9-2-12-28. (in Russian)
12. Korotkij V.A. Formoobrazovanie linij i poverhnostej na osnove krivyh vtorogo poryadka v komp'yuternom geometricheskom modelirovanii. Dokt. Diss. 05.01.01 [Shaping of lines and surfaces based on second-order curves in computer geometric modeling. Doct. Diss. 05.01.01]. Nizhnij Novgorod, 2018. 38 p. (in Russian)
13. Panchuk K.L., YUrkov V.YU., Kajgorodceva N.V. Matematicheskie osnovy geometricheskogo modelirovaniya krivyh linij [Mathematical foundations of geometric modeling of curved lines]. Omsk, OmGTU Publ., 2020, 198 p. (in Russian)
14. Peklich V.A. Vysshaya nachertatel'naya geometriya [Higher Descriptive Geometry]. Moscow, ASB Publ., 2000. 344 p. (in Russian)
15. Rashevskij P.K. Kurs differencial'noj geometrii uchebnik dlya gosudarstvennyh universitetov [Differential Geometry Course: Textbook for public universities]. Moscow, LKI Publ., 2008. 428 p. (in Russian)
16. Savelov A.A. Ploskie krivye. Sistematika, svojstva, primeneniya [Flat curves. Systematics, properties, applications]. Moscow, Librokom Publ., 2014. 294 p. (in Russian)
17. Sal'kov N.A. Ob odnom sposobe formirovaniya konik [About one way of forming conics]. Geometriya i grafika [Geometry and Graphics]. 2022, V. 10, I. 4, pp. 3-12. DOI:https://doi.org/10.12737/2308-4898-2022-10-4-3-12. (in Russian)
18. Smogorzhevskij A.S., Stolova E.S. Spravochnik po teorii ploskih krivyh tret'ego poryadka [Handbook of the theory of plane curves of the third order]. Moscow, Fizmatgiz Publ., 1961. 263 p. (in Russian)
19. Sogomonyan K.A. Linejno-konstruktivnye metody formoobrazovaniya (geometricheskoe modelirovanie) [Linear-constructive methods of shaping (geometric modeling)] Erevan, Ajastan Publ., 1990. 214 p. (in Russian)
20. Suncov O.S., ZHiharev L.A. Issledovanie otrazheniya ot krivolinejnyh zerkal na ploskosti v programme Wolfram Mathematica [Investigation of reflection from curved mirrors on a plane in the Wolfram Mathematica program]. Geometriya i grafika [Geometry and Graphics]. 2021, V. 9, I. 2. pp. 29-45. DOI:https://doi.org/10.12737/2308-4898-2021-9-2-29-45. (in Russian)
21. Sycheva A.A. Funkcional'no-voksel'noe modelirovanie krivyh Bez'e [Functional voxel modeling of Bezier curves]. Geometriya i grafika [Geometry and Graphics]. 2021, V. 9, I 4, pp. 63-72. DOI:https://doi.org/10.12737/2308-4898-2022-9-4-63-72. (in Russian)
22. Beglov I.A. Application of quasi-rotation surface segments in architectural prototyping / I.A. Beglov, V.V. Rustamyan and R.A. Verbitskiy // Journal of Physics: conference series, 15, Omsk, 9-11 Novembre 2021. Omsk, 2022. P. 012002. DOI:https://doi.org/10.1088/1742-6596/2182/1/012002.
23. Beglov I.A. Computer geometric modeling of quasi-rotation surfaces / I.A. Beglov. // Journal of physics: conference series: 5. Omsk, 16-17 March 2021. Omsk, 2021. P. 012057. DOIhttps://doi.org/10.1088/1742-6596/1901/1/012057.
24. Beglov I.A. Generation of the surfaces via quasi-rotation of higher order / I.A. Beglov. // Journal of physics: conference series: IV International Scientific and Technical Conference «Mechanical Science and Technology Update», MSTU 2020, Omsk, 17-19 March 2020. Omsk: Institute of physics publishing, 2020. P. 012032. DOIhttps://doi.org/10.1088/1742-6596/1546/1/012032.
25. Beglov I.A. N-n-digit interrelations between the sets within the R 2 plane generated by quasi-rotation of R 3 space / I.A. Beglov. // Journal of physics: conference series: IV International Scientific and Technical Conference «Mechanical Science and Technology Update», MSTU 2020, Omsk, 17-19 March 2020. Omsk: Institute of physics publishing, 2020. P. 012033. DOIhttps://doi.org/10.1088/1742-6596/1546/1/012033
26. Beglov I.A. Plane tangent to quasi-rotation surface / I.A. Beglov, K.L. Panchuk. // CEUR Workshop Proceedings: 30, Saint Petersburg, 22-25 September 2020. Saint Petersburg, 2020.
27. Panchuk K.L. Spatial spline construction through the Monge model / K.L. Panchuk, T.M. Myasoedova, Yu.A. Rogoza. // CEUR Workshop Proceedings: 30, Saint Petersburg, 22-25 September 2020. Saint Petersburg, 2020. DOIhttps://doi.org/10.51130/graphicon-2020-2-3-60.
28. Panchuk K.L. Spline curves formation given extreme derivatives / K.L. Panchuk, T.M. Myasoedova, E.V. Lyubchinov. // Mathematics. 2021. V. 9 (1). P. 1-29. DOIhttps://doi.org/10.3390/math9010047.
