Journal of Applied Mathematics

Surfaces Modelling Using Isotropic Fractional-Rational Curves

Igor V. Andrianov, Nataliia M. Ausheva, Yuliia B. Olevska, and Viktor I. Olevskyi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The problem of building a smooth surface containing given points or curves is actual due to development of industry and computer technology. Previously used for those purposes, shells of zero Gaussian curvature and minimal surfaces based on isotropic analytic curves are restricted in their consumer properties. To expand the possibilities regarding the shaping of surfaces we propose the method of constructing surfaces based on isotropic fractional-rational curves. The surfaces are built using flat isothermal and orthogonal grids and on the basis of the Weierstrass method. In the latter case, the surfaces are minimal. Examples of surfaces that were built according to the proposed method are given.

Article information

Source
J. Appl. Math., Volume 2019 (2019), Article ID 5072676, 13 pages.

Dates
Received: 30 April 2019
Accepted: 11 July 2019
First available in Project Euclid: 19 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.jam/1568858739

Digital Object Identifier
doi:10.1155/2019/5072676

Mathematical Reviews number (MathSciNet)
MR3993472

Citation

Andrianov, Igor V.; Ausheva, Nataliia M.; Olevska, Yuliia B.; Olevskyi, Viktor I. Surfaces Modelling Using Isotropic Fractional-Rational Curves. J. Appl. Math. 2019 (2019), Article ID 5072676, 13 pages. doi:10.1155/2019/5072676. https://projecteuclid.org/euclid.jam/1568858739


Export citation

References

  • Q. Li, Y. Su, Y. Wu, A. Borgart, and J. G. Rots, “Form-finding of shell structures generated from physical models,” International Journal of Space Structures, vol. 32, no. 1, pp. 11–33, 2017.
  • Y. H. Kim, M. Wu, and K. Kim, “Stress analysis of osteoporotic lumbar vertebra using finite element model with microscaled beam-shell trabecular-cortical structure,” Journal of Applied Mathematics, vol. 2013, Article ID 285165, 6 pages, 2013.
  • V. Olevskyi and Y. Olevska, “Mathematical model of elastic closed flexible shells with nonlocal shape deviations,” Journal of Geometry and Symmetry in Physics, vol. 50, pp. 57–69, 2018.
  • Y. Olevska, V. Mishchenko, and V. Olevskyi, “Mathematical models of magnetite desliming for automated quality control systems,” AIP Conference Proceedings, vol. 1773, Article ID 040007, 2016.
  • V. I. Mossakovskii, A. M. Mil'tsyn, and V. I. Olevskii, “Deformation and stability of technologically imperfect cylindrical shells in a nonuniform stress state,” International Journal of Strength of Materials, vol. 22, no. 12, pp. 1745–1750, 1990.
  • W. Blaschke, Vorlesungen über Differentialgeometrie Und Geometrische Grundlagen Von Einsteins Relativitätstheorie, vol. 1, Elementare Differentialgeometrie, Springer, Berlin, Germany, 1921.
  • E. O. Chernyshova, Using the functions of a complex variable for the construction of surfaces of technical forms PhD thesis's, KNUBA, Kyiv, 2007.
  • I. O. Korovina, Construction of surfaces of a stable mean curvature by given lines of an incident, author's abstract PhD thesis's, KNUBA, Kyiv, 2012.
  • S. F. Pilipaka and I. O. Korovin, “Construction of the minimum surface by screw motion of the spatial curve. Applied geometry and engineering graphics,” Tavria State Agrotechnological University, vol. 39, no. 4, pp. 30–36, 2008.
  • E. Cartan, The Theory of Finite Continuous Groups and Differential Geometry, Set Out by The Method of A Movable Rapper, Moscow University, 1963.
  • Z. Wang, D. Pei, L. Chen, L. Kong, and Q. Han, “Singularities of focal surfaces of null cartan curves in minkowski 3-space,” Abstract and Applied Analysis, vol. 2012, Article ID 823809, 20 pages, 2012.
  • Ü. Pekmen, “On minimal space curves in the sense of Bertrand curves,” University of Belgrade Publications of Electrotechnical Faculty. Serie Mathematics, no. 10, pp. 3–8, 1999.
  • J.-M. Hwang, “Geometry of minimal rational curves on Fano manifolds,” in School on Vanishing Theorems and Effective Results in Algebraic Geometry, vol. 6, pp. 335–393, Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, 2000.
  • J. Beban-Brkic and V. Volenec, “Butterflies in the isotropic plane,” KoG, vol. 8, no. 8, pp. 29–35, 2004.
  • B. Andrews, “Classification of limiting shapes for isotropic curve flows,” Journal of the American Mathematical Society, vol. 16, no. 2, pp. 443–460, 2003.
  • S. Y\ilmaz and M. Turgut, “Some characterizations of isotropic curves in the Euclidean space,” International Journal of Computational and Mathematical Sciences, vol. 2, no. 2, pp. 107–109, 2008.
  • N. M. Ausheva and A. A. Demchyshyn, “Construction of surfaces with orthogonal coordinate grids on the basis of isotropic curves,” Interdepartmental Scientific and Technical Collection: Applied Geometry and Engineering Graphics, vol. 91, pp. 2–7, 2013.
  • N. M. Ausheva, “Modeling of flat grids on the basis of fractional-rational isotropic curves,” Technological Audit and Production Reserves, vol. 6, no. 14, pp. 41–43, 2013.
  • N. M. Ausheva and G. A. Ausheva, “Isotropic fractional-rational curves of the third order,” Applied Geometry and Engineering Graphics. Proceedings Tavria State Agrotechnological University, vol. 4, no. 53, pp. 3–6, 2012.
  • N. M. Ausheva, “Isotropic polygons of isotropic Bezier curves,” Interdepartmental Scientific and Technical Collection: Applied Geometry and Engineering Graphics, vol. 88, pp. 57–61, 2011.
  • N. M. Ausheva and A. A. Demchyshyn, “Bending of minimal surfaces in a complex space of deformation of the Bézier curve,” Interdepartmental Scientific and Technical Collection: Applied Geometry and Engineering Graphics, vol. 90, pp. 15–19, 2012.
  • V. V. Dzyuba, Construction and transformation of surfaces with preservation of lines of curvature, author's abstract [PhD thesis's], KNUBA, Kyiv, Ukraine, 2008.
  • S. M. Kovalev and O. V. Vorontsov, “Construction of mesh frames of surfaces from the horizontal and the lines of the greatest inclination,” Applied Geometry and Engineering Graphics, vol. 54, pp. 13–16, 1993. \endinput