Study of the problem on constructing quadrics at the assigned tangent cones

Authors

DOI:

https://doi.org/10.15587/1729-4061.2019.180859

Keywords:

quadric, tangent cones, surface determinant, contact line, perspective image, conjugation of surfaces

Abstract

The research focuses on solving the problems, related to modeling the second-order surfaces (quadrics), the determinant of which includes tangent cones. All research was performed by the paradigm of using constructive methods of creation of algorithms. This is caused by the fact that there is a possibility to rely on a significant number of basic geometric tasks implemented in the DSS.

The problem of modeling of quadrics by tangent cones is relevant because there are at least its two important applications. The first is the construction of the surfaces by its contour line on perspective images. In this case, the point of view and the perspective contour line assign the enveloping cone, which in the case of quadrics coincides with the tangent cone. Such problems are solved in the context of problems of technical aesthetics and architectural design.

The second application is found in the problems of constructing the quadrics that are conjugated by the assigned curves or in the problems of conjugation of two quadrics by the third one. The problem of conjugation of surfaces is of wide practical significance, which is proved by the interest of users and developers of computer modeling systems.

Within the framework of this research, the existing theoretical geometric properties for modeling the quadrics, the determinant of which includes tangent cones, were accumulated, and a series of new geometric properties were found.

We developed the method, by which through assigning the contact line on one cone, there is a possibility to find a contact line on the second cone, as well as to find the center of the quadric inscribed in these two cones. An alternative method for modeling the described surfaces was also proposed. In this way, the cross-sections of all quadrics, tangent to two cones, are inscribed in quadrangulars, the vertices of which belong to the lines of the intersection of the assigned cones. Based of structural geometric research, the algorithms for computer realization of problems of modeling objects by their contour lines on their perspective images were developed.

The research results in the form of the theoretical calculations and examples of their application show the effectiveness of the proposed algorithms. The described approach to the solution of the stated tasks makes it possible to extend the possibilities of existing computer systems in their being applied in the work of designers and greatly simplify the process of the creation of actual objects

Author Biographies

Vira Аnpilogova, Kyiv National University of Construction and Architecture Povitroflotsky ave., 31, Kyiv, Ukraine, 03680

PhD, Professor

Department of Descriptive Geometry and Engineering Graphics

Svitlana Botvinovska, Kyiv National University of Construction and Architecture Povitroflotsky ave., 31, Kyiv, Ukraine, 03680

Doctor of Technical Sciences, Associate Professor

Department of Descriptive Geometry and Engineering Graphics

Alla Zolotova, Kyiv National University of Construction and Architecture Povitroflotsky ave., 31, Kyiv, Ukraine, 03680

PhD

Department of Descriptive Geometry and Engineering Graphics

Hanna Sulimenko, Zhytomyr National Agroecological University Staryi blvd., 7, Zhytomyr, Ukraine, 10008

PhD, Associate Professor

Department of Computer Technologies and Systems Modeling

References

  1. Gfrerrer, A., Zsombor-Murray, P. (2009). Quadrics of Revolution on Given Points. Journal for Geometry and Graphics, 13 (2), 131–144.
  2. Zsombor-Murray, P., Fashny, S. (2006). A Cylinder of Revolution on Five Points. Journal for Geometry and Graphics, 10 (2), 207–213.
  3. Breuils, S., Nozick, V., Sugimoto, A., Hitzer, E. (2018). Quadric Conformal Geometric Algebra of R9,6. Advances in Applied Clifford Algebras, 28 (2). doi: https://doi.org/10.1007/s00006-018-0851-1
  4. Emery, J. Conics, Quadrics and Projective Space. Available at: http://www.stem2.org/je/quadric.pdf
  5. Trocado, A., Gonzalez-Vega, L. (2019). Computing the intersection of two quadrics through projection and lifting. Available at: https://arxiv.org/pdf/1903.06983.pdf
  6. Bobenko, A. I., Suris, Yu. B. Discrete differential geometry. Consistency as integrability. Available at: https://arxiv.org/pdf/math/0504358.pdf
  7. Doliwa, A. (1999). Quadratic reductions of quadrilateral lattices. Journal of Geometry and Physics, 30 (2), 169–186. doi: https://doi.org/10.1016/s0393-0440(98)00053-9
  8. Polezhaev, Yu. O., Fatkullina, A. A., Borisova, A. Yu. (2012). Geometric models of junctions of quadrics in fragments of architectural pieces. Vestnik MGSU, 9, 18–23.
  9. Mihaylenko, V. E., Obuhova, V. S., Podgorniy, A. S. (1972). Formoobrazovanie obolochek v arhitekture. Kyiv: Budіvel'nik, 207.
  10. SolidWorks. Available at: https://en.wikipedia.org/wiki/SolidWorks
  11. Korotkiy, V. A., Usmanova, E. A., Khmarova, L. I. (2017). Geometric Modeling of Construction Communications with Specified Dynamic Properties. IOP Conference Series: Materials Science and Engineering, 262, 012110. doi: https://doi.org/10.1088/1757-899x/262/1/012110
  12. Korotkiy, V. A., Usmanova, E. A., Khmarova, L. I. (2016). Dinamic connection of second-order curves. 2016 2nd International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM). doi: https://doi.org/10.1109/icieam.2016.7911687
  13. Korotkiy, V. A. (2018). Construction of a Nine-Point Quadric Surface. Journal for Geometry and Graphics, 22 (2), 183–193.
  14. Heyfets, A. L. (2002). Issledovanie linii peresecheniya poverhnostey vtorogo poryadka v kurse teoreticheskih osnov komp'yuternogo geometricheskogo modelirovaniya. International Conference Graphicon 2002. Available at: http://graphicon2002.unn.ru/demo/2002/Kheyfets_2_Re.pdf
  15. Ivanov, G. S. (2014). Konstruktivniy sposob issledovaniya svoystv parametricheski zadannyh krivyh. Geometriya i grafika, 2 (3), 3–6.
  16. Ivanov, G. S., Zhirnyh, B. G. (2015). Geometricheskoe obespechenie postroeniya gladkih sopryazheniy iz otsekov konicheskih poverhnostey vtorogo poryadka. Inzhenerniy vestnik, 06, 1026–1033. Available at: http://engsi.ru/doc/777408.html
  17. Monzh, G. (2013). Nachertatel'naya geometriya. Klassiki nauki. Moscow: Kniga po trebovaniyu, 292.
  18. Rahmann, S. (2003). Reconstruction of Quadrics from Two Polarization Views. Pattern Recognition and Image Analysis, 810–820. doi: https://doi.org/10.1007/978-3-540-44871-6_94
  19. Sazonov, K. A., Yankovskaya, L. S. (2009). Komp'yuternoe modelirovanie sfericheskih poverhnostey obektov dizayna na perspektivnyh izobrazheniyah. Tekhnichna estetyka i dyzain, 6, 19–26.
  20. Sulimenko, S. Yu., Anpilohova, V. O, Levina, Zh. H. (2016). Formoutvorennia poverkhon obertannia druhoho poriadku za yikh liniyamy obrysiv. Suchasni problemy arkhitektury ta mistobuduvannia, 44, 320–325.
  21. Sulimenko, S. Yu. (2016). Konstruktyvno-parametrychnyi analiz formoutvorennia elipsoidiv za yikh liniyamy obrysiv. Suchasni problemy arkhitektury ta mistobuduvannia, 42, 109–115.
  22. Modenov, P. S. (1969). Analiticheskaya geometriya. Moscow: MGU, 688.
  23. Adamar, Zh. S. (1951). Elementarnaya geometriya. Chast' vtoraya. Stereometriya. Moscow: ONTI, 760.

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Published

2019-10-16

How to Cite

Аnpilogova V., Botvinovska, S., Zolotova, A., & Sulimenko, H. (2019). Study of the problem on constructing quadrics at the assigned tangent cones. Eastern-European Journal of Enterprise Technologies, 5(1 (101), 39–48. https://doi.org/10.15587/1729-4061.2019.180859

Issue

Section

Engineering technological systems