Our website uses cookies. By continuing to use this site, you agree to our use of cookies in accordance with this notice and the Terms of Use.
Journals Monographies Textbooks Collections Authors
Submit manuscript
Geometry & Graphics

ON THE INTERCONNECTION OF A RANGE AND THE SECOND-ORDER CLUSTER On the Interconnection of a Range and the Second-Order Cluster

Published in Geometry & Graphics · Volume 4, Issue 2 · Pages 8–18 · Rubric: Scientific problems of geometry
DOI: https://doi.org/10.12737/19828
Received: 25.05.2016 Accepted: 17.06.2016 Published: 17.06.2016 Language of publication: ru-RU
Authors
O.
O. Grafskiy
ABSTRACT
In accordance with “Specialized sections of affine, projective and computational geometry” syllabus for Master’s degree program in “Multimedia systems and computer graphics” developed at the Far Eastern State Transport University, the subject “Projective theory of the second-order curves” is considered [4; 14; 18]. Both at the sources mentioned and the textbook [11] projective method of the second-order curves formation as a range of the second order and its dual form – a second-order cluster (with regard to well-known theorems and relations, including Pascal and Brianchon theorems) is discernible. However, the graphical interpretations represented at the sources mentioned have general abstract character: to form the secondorder range two projective clusters of the first-order with the corresponding right lines are defined, and to design the second-order range – two projective series with the corresponding points. Techniques of high value can be observed when constructing outlines with the second-order curves; in this case, depending on engineering discriminant values, these curves can be constructed both using Pascal lines and qualities of the engineering discriminant itself, that is paying attention to the fact that tangents to the second-order curves makes the second-order cluster. Naturally, intent arises not to set the corresponding points on projective ranges, but to get them by elaboration, disclosing upon that regularities when constructing different second-order curves (the first aspect of research). The second aspect is in the consider - ation of the particular cases which would have definite secondorder clusters. In this case the task would be to model the secondorder range as a dual form of cluster. Thus it would be possible to get the interconnection of the definite cluster and the second-order cluster.
KEYWORDS
second-order curves and ranges, Pascal line, engineering discriminant, trigonometric functions of complex variable, visualization tools in Maple package.
Text References
Text

В соответствии с программой дисциплины «Спецразделы аффинной, проективной и вычислительной геометрии» для подготовки магистров по профилю «Системы мультимедиа и компьютерная графика» при ДВГУПС, рассматривается тема «Проективная теория кривых второго порядка» [4; 14; 18]. В указанных источниках, а также учебном пособии [11], прослеживается проективный способ образования кривых второго порядка как ряда второго порядка, а также двойственная его форма — пучок второго порядка (принимая во внимание известные теоремы, следствия, включая теоремы Паскаля и Брианшона).

Однако представленные графические интерпретации в указанных выше источниках имеют общий теоретический характер: для построения ряда второго порядка задаются два проективных пучка первого порядка с соответственными прямыми, а при конструировании пучка второго порядка — два проективных ряда с соответственными точками. Более значимые приемы можно наблюдать при построении обводов кривыми второго порядка: здесь, в зависимости от значений инженерного дискриминанта, можно строить эти кривые как при помощи прямых Паскаля, так и используя свойство самого инженерного дискриминанта, т.е. принимая во внимание, что проводимые касательные к кривым второго порядка и составляют пучок второго порядка.

1. Grafskiy O.A. Geometricheskiy analiz algebraicheskikh krivykh v ploskosti [Geometric analysis of algebraic curves on a plane. А research report]. Khabarovsk, 2009. 86 p.

2. Girsh A.G. Mnimosti v geometrii [Imaginaries in Geometry]. Geometriya i grafika [Geometry and Graphics]. V. 2, I. 2, pp 3-8. (in Russian). DOI:https://doi.org/10.12737/14415.

3. Girsh A.G. Fokusy algebraicheskikh krivykh [Focuses of algebraic curves]. Geometriya i grafika [Geometry and Graphics]. V. 3, I. 3, pp 4-17. (in Russian). DOI:https://doi.org/10.12737/14415.

4. Glagolev N.A. Proektivnaya geometriya [Projective geometry]. Moscow, Vysshaya shkola Publ., 1963. 334 p.

5. Grafskiy O.A., Doronina S.S., Galliulin N.Kh. Analiz postroeniya krivykh vtorogo poryadka [Analysis of the construction of second-order curves]. Nauchno-tekhnicheskoe i ekonomicheskoe sotrudnichestvo stran ATR v XXI veke: Materialy Vserossiyskoy nauchno-prakticheskoy konf. s mezhdunarodnym uchastiem, 22-24 aprelya 2009 g. [Scientific and technical and economic cooperation between Asia-Pacific countries in the XXI century: Proceedings of All-Russian Scientific-Practical Conference. with international participation, 22-24 April 2009]. Khabarovsk, DVGUPS Publ., 2009, V. 6, pp. 165-168. (in Russian).

6. Grafskiy O.A, Saenko O.V. Kasatel´naya k okruzhnosti [The tangent to the circle]. Nauchno-tekhnicheskie problemy transporta, promyshlennosti i obrazovaniya: tr. Vseros. nauch.-prakt. konf. [Scientific and technical problems of transport, industry and education: the works of the All-Russian scientific-practical conference]. Khabarovsk, DVGUPS Publ., 2010. V. 6, pp. 190-192. (in Russian).

7. Grafskiy O.A, Galliulin N.Kh. K voprosu obosnovaniya konstruirovaniya ryada vtorogo poryadka [On the question of justification of the construction of a number of secondorder]. Sovremennye problemy i puti ikh resheniya v nauke, transporte, proizvodstve i obrazovanii 2008: Materialy mezhdunarodnoy nauchno-prakticheskoy Internet-konferentsii 15-25 dekabrya 2008 g. [Modern problems and their solutions in science, transport, manufacturing and education, 2008: Proceedings of the International scientific and practical Internet-conference on December 15-25, 2008.]. Odessa, Chernomor´e Publ, pp. 59-63. (in Russian).

8. Grafskiy O.A., Nasonova N.A. K voprosu postroeniya kasatel´noy k giperbole [The problem of constructing a tangent to the hyperbola]. Nauchno-tekhnicheskoe i ekonomicheskoe sotrudnichestvo stran ATR v XXI veke: tr. Vseros. molodezhnoy nauch.-prakt. konf. s mezhdunar. uchastiem 20-22 aprelya 2011 g. [Scientific and technical and economic cooperation between Asia-Pacific countries in the XXI century: Russian youth learn how to work and practical conference with participation international April 20-22, 2011]. Khabarovsk, DVGUPS Publ., 2011, V. 5, pp. 205-209. (in Russian).

9. Grafskiy O.A. Modelirovanie mnimykh elementov na ploskosti [Simulation imaginary elements in the plane]. Khabarovsk, DVGUPS Publ., 2004. 161 p.

10. Grafskiy O.A., Saenko O.V. Obosnovanie postroeniya kasatel´noy k okruzhnosti i ellipsu [Rationale for constructing a tangent to the circle and ellipse]. Nauchno-tekhnicheskoe i ekonomicheskoe sotrudnichestvo stran ATR v XXI veke: tr. Vseros. molodezhnoy nauch.-prakt. konf. s mezhdunar. uchastiem 20-22 aprelya 2011 g. [Scientific and technical and economic cooperation between Asia-Pacific countries in the XXI century: labor Vserossosiyskoy youth scientific and praktticheskoy conference with international participation 20-22 April 2011]. Khabarovsk, DVGUPS Publ., V. 5, pp. 209-211. (in Russian).

11. Grafskiy O.A. Fundamentals of affine and projective geometry. Tutorial. Khabarovsk, DVGUPS Publ., 2013. 135 p.

12. Grafskiy O.A. Osnovy affinnoy i proektivnoy geometrii. Kand. Diss. [Theoretical and constructive problems of modeling of imaginary elements in descriptive geometry and its applications. Cand. Diss.]. Moscow, 2004. 406 p.

13. Ivanov G.S. Teoretiko-konstruktivnye problemy modelirovaniya mnimykh elementov v nachertatel´noy geometrii i ee prilozheniyakh [Theoretical foundations of descriptive geometry]. Moscow, Mashinostroenie Publ. 1998. 157 p.

14. Ivanov G.S., Dmitrieva I.M. Teoreticheskie osnovy nachertatel´noy geometrii [On the tasks of descriptive geometry with imaginary solutions]. Geometriya i grafika [Geometry and Graphics]. 2015, no. 2. Pp. 3-8. (in Russian). DOI:https://doi.org/10.12737/12163.

15. Grafskiy O.A. Innovatsii pri izuchenii studentami proektivnoy geometrii/Innovatsii v teorii geometricheskogo modelirovaniya pri izuchenii studentami tekhnicheskikh vuzov fundamental´nykh i spetsial´nykh distsiplin [Innovations in the study of projective geometry by students / Innovation in geometric modeling theory in the study of students of technical colleges the basic and special disciplines]. Khabarovsk, 2012. 106 p.

16. Grafskiy O.A. Otobrazheniya, preobrazovaniya i geometricheskiy analiz ploskikh algebraicheskikh krivykh / Innovatsii v teorii geometricheskogo modelirovaniya pri izuchenii studentami tekhnicheskikh vuzov fundamental´nykh i spetsial´nykh distsiplin [The mappings, transformations, and geometric analysis of plane algebraic curves / Innovation in geometric modeling theory in the study of students of technical colleges the basic and special disciplines]. Khabarovsk, VNTITsentr Publ., 2010, 73 p.

17. Sal´kov N.A. Ellips: kasatel´naya i normal´ [Ellipse: tangent and normal]. Geometriya i grafika [Geometry and Graphics]. 2013, I. 1, pp. 35-37. (in Russian). DOI:https://doi.org/10.12737/2084.

18. Chetverukhin N.F. Proektivnaya geometriya [Projective geometry]. Moscow, Prosveshchenie Publ., 1969. 386 p.

JATS XML
The contact persons
Zamestitel' glavnogo redaktora, chlen redsoveta
Vladimir Vyshnyepolskiy
Glavnyy redaktor
A. A. Ishchenko
Vypuskayuschiy redaktor
Dar'ya Sklyankina
501@infra-m.ru
Deputy editor-in-chief, member of the editorial Board
N. Sal'kov
nikolaysalkov@mail.ru
RINC Neopoisk
© INFRA-M
Agreement Personal data protection and processing policy Powered by Editorum, 2026
Login or Create
* Forgot your password?