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Geometry & Graphics
UDC
51

PROPERTIES FEATURES OF PARABOLA AT ITS SIMULATION Properties Features of Parabola at Its Simulation

Published in Geometry & Graphics · Pages 63–77 · Rubric: Scientific problems of geometry
DOI: https://doi.org/10.12737/article_5b55a16b547678.01517798
Received: 23.07.2018 Accepted: 21.08.2018 Published: 21.08.2018 Language of publication: RUS
Authors
O.
O. Grafskiy
YU
Yu. Ponomarchuk
V.
V. Suric
ABSTRACT
When studying the theory of contour construction in “Affine and Projective Geometry” course on educational program specializations “Computer-Aided Design Systems” and “Applied Informatics in Design” a unit of computational and graphic task "Contour Construction" is carrying out in structural design. In this computational and graphic task the contour constructions are carrying out by second-order curves (a circle — by the radius and graphical method; a hyperbola, an ellipse, a parabola — by means of Pascal curves, taking into account positions of engineering discriminant). The constructions of an arc of ellipse, hyperbola, and parabola are carried out based on Pascal theorem: in any hexagon, which vertices belong to a second-order series, three points of the opposite sides’ intersection lie on one straight line — the Pascal line. However, in construction of a conic (a second-order curve), it is necessary to draw students’ attention to the fact that the points belonging to a second-order series (a second-order curve, or a conic) make a geometrical locus of intersection of Pascal hexagon’s adjacent opposite sides. By this method students successfully construct conjugate arcs of an ellipse and a hyperbola with other conics. The construction of a parabola arc, conjugated with other conics, is carried out by the method of engineering discriminant (it is more convenient to divide line segments in halves: a median and a triangle side, which is opposite to its vertex lying on a parabola arc). It should be noted that theoretical and practical material on this subject corresponds to the assimilation of Study Plan’s necessary competences (in accordance with each educational program), however, some aspects of this subject are accepted by students simply by trust. The aim of this paper is research of construction methods for parabola, applied to contour simulation.
KEYWORDS
parabola, Pascal line, engineering discriminant, Hermite curve’s segment, Bezout, B-spline.
Text References
Text

Плоская алгебраическая кривая n-го порядка имеет параметрическое число, равное



Для коники в соответствии с выражением (1) параметрическое число равно 5. Это объясняется и общим положением с позиции аналитической геометрии: исходя из уравнения кривой 2-го порядка в декартовых координатах
Ax2 + By2 + Cxy + Dx + Ey + F = 0 (2) имеем 6 коэффициентов (A, B, C, D, E, F). Если поделить каждый член уравнения (2) на коэффициент F, то уравнение (2) примет вид 

Ax2 + By2 + Cxy + Dx + Ey +1 = 0, в котором 5 параметров. Поэтому заключаем, что для ее задания необходимо 5 параметров — количество коэффициентов: A′, B′, C′, D′, E′, т.е. это ∞5 множество точек, или любые другие условия, сохраняющие именно 5 параметров: 5 точек; три точки и две касательные и т. д. [6; 12; 19]. Однако в случае, если коника проходит через начало координат, то в выражении (2) коэффициент F = 0 [3; 9; 14; 23]. Тогда уравнение такой кривой следует записать как
Ax2 + By2 + Cxy + Dx + Ey = 0, (3) т.е. 5 коэффициентов и, таким образом, 5 параметров. Коэффициенты A, B, C, D, E выражения (3) можно определить, подставив координаты точек в уравнение кривой. Получаем 5 уравнений первой степени. Решая систему пяти уравнений, узнаем искомые коэффициенты [5; 11; 14; 15; 19]. Приведем несколько примеров, применяемых как при конструктивном моделировании параболы (только при помощи прямых Паскаля и свойства инженерного дискриминанта, не затрагивая другие известные способы построения), так и с использованием средств компьютерной визуализации. Пример 1. Принимая «на веру», что при определении параболы и ее касательной t в точке A отрезки OAy и OAy равны, т.е. OA OA y y = (рис. 1), следует доказать это положение известной теоремы. Попутно приведем цитату: «Заметим синтез и анализ, не в математическом, а в общелогическом смысле слова совершенно равноправны, и во всяком исследовании они постоянно переплетаются друг с другом; поэтому едва ли может быть речь о предоставлении господства одному из этих орудий чело-
веческой мысли» [21]. В связи с этим проводимые исследования следует рассматривать и с аналитических позиций [18; 24; 25], которые обеспечивают моделирование с применением информационных технологий [10].

1. Girsh A.G. Mnimosti v geometrii [Imaginaries in Geometry]. Geometriya i grafika [Geometry and Graphics]. 2014, V. 2, I. 2, pp. 3-8. DOI:https://doi.org/10.12737/5583. (in Russian)

2. Glagolev N.A. Proektivnaja geometrija [Projective geometry]. Moscow, Vysshaya shkola Publ., 1963. 344 p. (in Russian)

3. Grafskij 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, FESTU Publ., 2009, V. 6, pp. 165-168. (in Russian)

4. Grafskij O.A. Affinnaya i proyektivnaya geometriya [Affine and Projective Geometry]. Khabarovsk, FESTU Publ., 2013. 27 p. (in Russian)

5. Grafskiy O.A., Smetanina V.V., Ni E.N. Vzaimnaya svyaz' ryada i puchka vtorogo poryadka na primere funktsii Zhukovskogo [The interrelation of a series and a second-order pencil by the example of Zhukovskii's function]. Nauchnyy vzglyad v budushcheye: Mezhdunarodnaya nauchno-prakticheskaya Internet-konferentsiya SWorld «Intellektual'nyy potentsial XXI veka ‘2016» [Scientific view of the future: International scientific and practical Internet conference SWorld «Intellectual potential of the XXI century '2016»]. Odessa: KUPRIYENKO S.V. Publ., V. 4, I. 4, 2016, pp. 70-77. (in Russian)

6. Grafskij O.A. Vychislitel'naya geometriya [Computational geometry]. Khabarovsk: FESTU Publ., 2014. 150 p. (in Russian)

7. Grafskij O.A., Saenko O.V. Vychislitel'naya geometriya [Computational geometry]. Khabarovsk: FESTU Publ., 2013. 21 p.

8. Grafskij O.A., Usmanov A.V., Holodilov A.A. Grafoanaliticheskie issledovaniya involyucii [Graphic-analytical researches of involution]. Geometriya i grafika [Geometry and Graphics]. 2017, V. 5, I. 1, pp. 3-11. DOI:https://doi.org/10.12737/25118. (in Russian)

9. Grafskij 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)

10. Grafskij O.A., Ponomarchuk Yu.V. Nekotorye metodicheskie aspekty geometrograficheskoy podgotovki studentov [Some methodical aspects of geometrografichesky training of students]. Problemy i perspektivy razvitiya obrazovaniya v tehnicheskih vuzah [Problems and prospects of a development of education in technical colleges]. Khabarovsk, FESTU Publ., 2016, pp. 200-204. (in Russian)

11. Grafskij O.A. Ob ustanovlenii vzaimnoj svyazi ryada i puchka vtorogo poryadka [About establishment of an interconnection of a row and bunch of the second order]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 4, I. 2, pp. 8-18. DOI:https://doi.org/10.12737/19828. (in Russian)

12. Grafskij O.A. Osnovy affinnoy i proektivnoy geometrii [Basics of affine and projective geometry]. Khabarovsk: FESTU Publ., 2013. 135 p.

13. Grafskij O.A., Smetanina V.V, Ni E.N. Osobennosti krivykh Bez'ye i V-splaynov [Features of Bezier and B-spline curves]. Voprosy nauki i obrazovaniya: teoreticheskiye i prakticheskiye aspekty: Materialy Mezhdunarodnoy nauchno-prakticheskoy konferentsii 16 maya 2017 g. (Praga, Chekhiya): Vydavatel "Osviceni", NITS "Mir nauki" [Questions of science and education: theoretical and practical aspects: Proceedings of the International Scientific and Practical Conference May 16, 2017 (Prague, Czech Republic): Vydavatel "Osviceni", SRC "World of Science "]. 2017, pp. 99-106. (in Russian)

14. Grafskii O.A. Teoretiko-konstruktivnye problemy modelirovaniya mnimykh elementov v nachertatel'noy geometrii i ee prilozheniyakh. Dokt. Diss. [Teoretiko-construktive problems of modeling of imaginary elements in descriptive geometry and its applications. Doct. Diss.]. Moscow, 2004. 406 p. (in Russian)

15. Ivanov G.S. Nachertatel'naya geometriya [Descriptive geometry]. Moscow, Mashinostroyeniye Publ., 1995. 224 p. (in Russian)

16. Ivanov G.S., Dmitriyeva I.M. Nelineynyye formy v inzhenernoy grafike [Nonlinear forms in engineering graphics]. Geometriya i grafika [Geometry and Graphics]. 2017, V. 5. I. 2. pp. 30-41. (in Russian)

17. Ivanov G.S, Dmitrieva, I.M. O zadachah nachertatelnoy geometrii s mnimymi resheniyami [About the Tasks of Descriptive Geometry With Imaginary Solutions]. Geometriya i grafika [Geometry and Graphics]. 2015, V. 3, I. 2, pp. 3-8. DOI:https://doi.org/10.12737/12163. (in Russian)

18. Ivanov G.S., Dmitrieva I.M. Princip dvojstvennosti - teoreticheskaya baza vzaimosvyazi sinteticheskih i analiticheskih sposobov resheniya geometricheskih zadach [The principle of a duality - theoretical base of interrelation of synthetic and analytical ways of the solution of geometrical tasks]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 4, I. 3, pp. 3-10. DOI:https://doi.org/10.12737/21528. (in Russian)

19. Ivanov G.S. Teoreticheskie osnovy nachertatel'noy geometrii [Theoretical fundamentals of descriptive geometry]. Moscow, Mashinostroenie Publ., 1998. 157 p. (in Russian)

20. Li K. Osnovy SAPR (CAD/CAM/CAE) [Fundamentals of CAD (CAD / CAM / CAE)]. St. Petersburg, Peter Publ., 2004. 560 p. (in Russian)

21. Protokol 79-go zasedanija. 31 marta 1898 g. [The Protocol of the 79th meeting. 31 Mar, 1898]. Izvestija Fiziko-matematicheskogo obshhestva Imperatorskogo Kazanskogo universiteta [News of Physical and mathematical society of Imperial Kazan university] Kazan, Tipolitografiya of the Imperial university Publ., 1898, V. VIII, I. 2, pp. 18-20. (in Russian)

22. Rodzhers D. Matematicheskiye osnovy mashinnoy grafiki [Mathematical foundations of computer graphics]. Moscow, Mir Publ., 1989. 512 p. (in Russian)

23. Savelov A.A. Ploskie krivye. Sistematika, svoystva, primeneniya [Flat curves. Systematization, properties, applications]. Moscow, RHD Publ., 2002. 294 p. (in Russian)

24. Sal´kov N.A. Nachertatel'naya geometriya - baza dlja geometrii analiticheskoj [Geometry As the Basis for Analytical Geometry]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 4, I. 1, pp. 44-54. DOI:https://doi.org/10.12737/18057. (in Russian)

25. Seregin V.I., Ivanov G.S., Dmitrieva I.M., Muravev K.A. Mezhdistsiplinarnye svyazi nachertatelnoy geometrii i smezhnyh razdelov vysshey matematiki [Interdisciplinary connections of descriptive geometry and related sections of higher mathematics]. Geometriya i grafika [Geometry and Graphics]. 2013, V. 1, I. 3/4, pp. 8-12. DOI:https://doi.org/10.12737/2124. (in Russian)

26. Stolbova I.D. Aktual'nyye problemy graficheskoy podgotovki studentov v tekhnicheskikh vuzakh [Actual problems of graphic preparation of students in technical universities]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 2, I. 1, pp. 30-41. DOI:https://doi.org/10.12737/3846. (in Russian)

27. Stolbova I.D. Ob obespechenii kachestva predmetnogo obucheniya studentov tehnicheskogo universiteta [Ensuring the Quality of Subject Teaching for Students at the Technical University]. Geometriya i grafika [Geometry and Graphics]. 2016, V. 3, I. 4, pp. 27-37. DOI:https://doi.org/10.12737/17348. (in Russian)

28. Chetveruhin N.F. Proektivnaja geometrija [Projective geometry]. Moscow, Prosveshhenie Publ, 1969. 368 p.

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