Construction of Belonging to Surfaces One-Dimensional Contours by Mapping Them to a Plane

Gennadiy S. Ivanov

doi:10.12737/article_5ad07ed61bc114.52669586

UDC 51

CONSTRUCTION OF BELONGING TO SURFACES ONE-DIMENSIONAL CONTOURS BY MAPPING THEM TO A PLANE Construction of Belonging to Surfaces One-Dimensional Contours by Mapping Them to a Plane

АА Аа

Published in Geometry & Graphics · Volume 6, Issue 1 · Pages 3–9 · Rubric: Scientific problems of geometry

DOI: https://doi.org/10.12737/article_5ad07ed61bc114.52669586

Received: 13.04.2018 Accepted: 24.03.2024 Language of publication: ru-RU

Authors

Gennadiy S. Ivanov

Academic degree: doctor of technical sciences

Bauman Moscow State Technical University Professor · Russian Federation · Moscow

ABSTRACT

As is known, differential geometry studies the properties of curve lines (tangent, curvature, torsion), surfaces (bending, first and second basic quadratic forms) and their families in small, that is, in the neighborhood of the point by means of differential calculus. Algebraic geometry studies properties of algebraic curves, surfaces, and algebraic varieties in general [1; 17]: order, class, genre, existence of singular points and lines, curves and surfaces family intersections (sheaves, bundles, congruences, complexes and their characteristics). Rational curves and surfaces occupy a special place among them: • their design by bi-rational (Cremona) transformations [10; 21]; • investigation of their properties by mapping to lines and planes [9; 21; 22]; • construction of smooth contours from arcs of rational curves belonging to surfaces [10]. It seems that the main results obtained in this direction by mathematicians in the second half of the 19th century by structural and geometric methods should be the theoretical support for the design of technical forms that meet a number of pre-set requirements using modern computational tools and information technologies. It is obvious that application of Cremona transformations’ powerful apparatus is useful when designing, for example, pipes of complex geometry according to set of streamlines, thin-walled shells for a given mesh manifold of curvature lines etc. Apparently, this stage should precede computer graphics’ calculation procedures. However, in Russian publications on applied (engineering) geometry, only a little attention is paid to the study of surfaces in general. The author knows nothing about the use of this approach for solving of these applied problems. In this regard, the aims of this paper are: • illustration of method for mapping a surface to a plane to study its properties in general by the example of construction a flat model for a hyperboloid of one sheet; • constructive approach to the construction of smooth one-dimensional contours on rational surfaces.

KEYWORDS

norm curve, hyperboloid of one sheet, mapping, stereographic, curved and oblique projection, Hirst transformation, one-dimensional contour.

Text

В первой половине XIX в. появились новые виды геометрии: проективная и алгебраическая, неевклидовы геометрии Лобачевского — Бойяи и Римана.

1. Aleksandrov A.D., Zalgaller V.A. Geometrija v celom [Geometry as a whole]. Matematicheskaja jenciklopedija [Mathematical encyclopedia]. Moscow, V. 1, 1977, pp. 943-944. (in Russian)

2. Bozhko A.N., Zhuk D.M., Manichev V.B. Komp'juternaja grafika [Computer graphics]. Moscow, Bauman Moscow State Technical University Publ., 2007. (in Russian)

3. Borovikov I.F., Ivanov G.S., Seregin V.I., Surkova N.G. Novye podhody prepodavanija nachertatel'noj geometrii v uslovijah ispol'zovanija informacionnyh obrazovatel'nyh tehnologij [New approaches of teaching descriptive geometry in the terms of use of educational information technology]. Inzhenernyy vestnik MGTU im. N.E. Baumana [Engineering Bulletin, Publishing house of Bauman Moscow State Technical University]. 2014, I. 12. (in Russian)

4. Glagolev N.A. Proektivnaja geometrija [Projective geometry]. Moscow, Vysshaja shk. Publ., 1963. 343 p. (in Russian)

5. Golovanov N.N. Geometricheskoe modelirovanie [Geometric modeling]. Moscow, Fizmatgiz Publ., 2002. (in Russian)

6. Guznenkov V.N., Jakunin V.I. Geometro-graficheskaja podgotovka kak integrirujushhij faktor obrazovatel'nogo processa [Geometro - graphic training as an integrating factor in the educational process.] Obrazovanie i obshhestvo [Education and society]. Moscow, 2014, I. 2, pp. 26-28. (in Russian)

7. Guznenkov V.N., Jakunin V.I. Principy formirovanija struktury i soderzhanija geometro-graficheskoj podgotovki [Principles of formation of the structure and content of geometric-graphic preparation]. Standarty i monitoring v obrazovanii [Standards and monitoring in education]. 2013, I. 6, pp. 34-39. (in Russian)

8. Efimov N.V. Neevklidovy geometrii [non-Euclidean geometry]. Matematicheskaja jenciklopedija [Mathematical encyclopedia]. Moscow, V. 3, 1982, pp. 910-914. (in Russian)

9. Ivanov G.S. Teoreticheskie osnovy nachertatel'noj geometrii [Theoretical foundations of descriptive geometry]. Moscow, Mashinostroenie Publ., 1988. (in Russian)

10. Ivanov G.S. Konstruirovanie tehnicheskih poverhnostej (matematicheskoe modelirovanie na osnove nelinejnyh preobrazovanij) [Designing of technical surfaces (mathematical modelling on the basis of nonlinear transformations)]. Moscow, Mashinostroenie Publ., 1987. 192 p. (in Russian)

11. Ivanov G.S. Normkrivaja trehmernogo prostranstva kak chastnyj sluchaj peresechenija dvuh kvadrik [Normalcy curve of three-dimensional space as a special case of intersection of two quadrics]. Trudy XXII mezhdunarodnoj nauchno-tehnicheskoj konferencii «Informacionnye sredstva i tehnologii» [Proceedings of the XXII international scientific and technical conference «Information means and technologies»]. Moscow, V. 2, MJeI Publ., 2014, pp. 51-56. (in Russian)

12. Ivanov G.S., Moskalenko V.O., Murav'ev K.A. Kak obespechit' obshhegeometricheskuju podgotovku studentov tehnicheskih universitetov [How to ensure to obseruations training of students in the technical universities]. Nauka i obrazovanie, MGTU im. N.E. Baumana [Science and education]. 2012, I. 8. Available at: http:// technomag.edu.ru/doc/445140.html (in Russian)

13. Ivanov G.S., Seregin V.I. Inzhenernaja geometrija - teoreticheskaja baza postroenija geometricheskih modelej [Engineering geometry - the theoretical basis for the construction of geometric models]. Sb. statej mezhdunarodnoj nauchno-prakticheskoj konferencii «Innovacionnoe razvitie sovremennoj nauki» [Collection of articles of the international scientific-practical conference «Innovative development of modern science»]. Ufa, BashGU Publ., 2014, pp. 339-346. (in Russian)

14. Konokbaev K.K. Konstruirovanie obvodov iz dug unikursal'nyh cirku-ljarnyh krivyh posredstvom kremonovyh involjucij. Kand. Diss. [Designing the contours of the arcs unicursal circular curves by Cremona of involutions. Cand. Diss.]. Moscow, MAI Publ., 1972, p. 21. (in Russian)

15. Miroljubova T.I. Geometricheskie modeli fasonnyh jelementov odnorukavnyh kanalovyh poverhnostej. Kand. Diss. [Geometric models of shaped elements of single-arm channel surfaces. Cand. Diss.]. Moscow, MAI, 2004, pp. 23. (in Russian)

16. Mul'dekov I.O. Reshenie konstruktivnyh zadach opisanija krivyh i po-verhnostej na osnove metodov optimizacii. Kand. Diss. [The solution of design problems of the description of curves and surfaces based on optimization techniques. Cand. Diss.]. Moscow, MGUPP Publ., 1996, p. 30. (in Russian)

17. Poznjak Je.G. Geometrija [Geometry]. Matematicheskaja jenciklopedija [Mathematical encyclopedia]. Moscow, 1977, V. 1, pp. 940-943. (in Russian)

18. Rozenfel'd B.A., I.M. Jaglom Neevklidovy geometrii [non-Euclidean geometry]. Jenciklopedija jelementarnoj matematiki [Encyclopedia of elementary mathematics]. Moscow, V. 5, 1966, pp. 394-476. (in Russian)

19. Seregin V.I., Ivanov G.S., Dmitrieva I.M., Murav'ev K.A. Mezhdis-ciplinarnye svjazi nachertatel'noj geometrii i smezhnyh razdelov vysshej matematiki [Interdisciplinary connections descriptive geometry and related sections of higher mathematics]. Geometrija i grafika [Geometry and graphics]. Moscow, INFRA-M Publ., 2013, V. 1, I. 3-4, pp. 8-12. - DOI:https://doi.org/10.12737/2124. (in Russian)

20. Foks A., Pratt M. Vychislitel'naja geometrija. [Computational geometry]. Moscow, Mir Publ., 1982. (in Russian)

21. Hudson H.P. Cremona transformation in plane and space. / H.P. Hudson. Cambridge. 1927. 454 p. (in English)

22. Semple J.G., Roth L. Introduction to algebraic geometry. Oxford, 1985. 480 p.

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Vladimir Vyshnyepolskiy

Glavnyy redaktor

A. A. Ishchenko

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Dar'ya Sklyankina

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Deputy editor-in-chief, member of the editorial Board

N. Sal'kov

nikolaysalkov@mail.ru

GOST

Ivanov G. S. Construction of Belonging to Surfaces One-Dimensional Contours by Mapping Them to a Plane // Geometry & Graphics. 2018. no. 1. pp. 3-9. DOI: https://doi.org/10.12737/article_5ad07ed61bc114.52669586. (Date of access 12.11.2026).

MLA

Gennadiy S. Ivanov. "Construction of Belonging to Surfaces One-Dimensional Contours by Mapping Them to a Plane." Geometry & Graphics 6.1 (2018): 3-9. DOI: https://doi.org/10.12737/article_5ad07ed61bc114.52669586.

APA

Ivanov G. S. (2018). Construction of Belonging to Surfaces One-Dimensional Contours by Mapping Them to a Plane. Geometry & Graphics, 6(1), 3-9. https://doi.org/10.12737/article_5ad07ed61bc114.52669586

Chicago

Gennadiy S. Ivanov. "Construction of Belonging to Surfaces One-Dimensional Contours by Mapping Them to a Plane." Geometry & Graphics 6, no. 1 (2018): 3-9. https://doi.org/10.12737/article_5ad07ed61bc114.52669586.

WIKI

{{cite journal | author = Ivanov G. S. | title = Construction of Belonging to Surfaces One-Dimensional Contours by Mapping Them to a Plane | journal = Geometry & Graphics | year = 2018 | volume = 6 | issue = 1 | pages = 3-9 | doi = 10.12737/article_5ad07ed61bc114.52669586 }}

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