TARU PUBLICATIONS
Journal of Dynamical Systems and Geometric Theories cover

Journal of Dynamical Systems and Geometric Theories

(Since 2002)
Open Access ·Peer-reviewed·ISSN (Online): 2169-0057·ISSN (Print): 1726-037X

WoS  JIF 2026 : 0.5 (Q3)

Powered by:Powered by

Half-Yearly Journal: Publishes research on dynamical systems and geometry, including Random Dynamical Systems, hyperbolic behaviour, complex geometry and Ergodic theory.

Issues up to 2022 co-published with and available at:Taylor & Francis
info@tarupublications.com
Taru Online First Current IssueBrowse all Issues
Journal of Dynamical Systems and Geometric Theories
Open Access Article

Differential geometry of transversal intersection curves of hypersurfaces in ℝ5

pp. 147–162Vol. 15Issue 2July 2017DOI: 10.1080/1726037X.2017.1411042XML
Received:
22 Nov 2016
Accepted:
13 Nov 2017
Published Online:
06 Dec 2017
Article type:
Article
Language:
EN
Article no.:
1726037X.2017.1411042
Pages:
147–162

Abstract

In this paper, we present the algorithms for calculating the differential geometric quantities {t, n, b1, b2, b3, k2, k3, k4}, geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space ℝ5. In transversal intersection, the normals of the surfaces at the intersection point are linearly independent, while as in nontransversal intersection, the normals of the surfaces at the intersection point axe linearly dependent.

Keywords

Hypersurfacestransversal intersectionnon-transversal intersection

Subject Classifications

35A0435A05

References

  1. N.H.Abdel-Alld, S. A. N.Badr, M.A.Soliman, S. A.Hassan, Intersection curves of hyper­surfaces in ℝ , Computer Aided Geometric Design 29 , 2012 , 99 – 108 .
  2. H.S.Abdel-Aziz, M.Khalifa Saad, A.A.Abdel-Salam, On implicit surfaces and their inter­section curves in Euclidean curves 4-spaces , Houston Journal of Mathematics , 40 ( 2 ), 2014 , 339 – 352 .
  3. O.Aléssio, Differential geometry of intersection curves in ℝ of three implicit surfaces . Comput. Aided Geom. Des. 26 ( 4 ), 2009 , 455 – 471 .
  4. O.Aléssio, Formulas for second curvature, third curvature, normal curvature, first geodesic curvature and first geodesic torsion of implicit curve in n-dimensions . Comput. Aided Geom. Des. , 29 ( 4 ), 2012 , 189 – 201 .
  5. O.Aléssio, Geometria diferencial de curvas de interseção de duas superficies implícitas . TEMA Tend. Mat. Apl. Comput. , 7 ( 2 ), 2006 , 169 – 178 .
  6. M. Düldül, Akbaba, Willmore-like methods for the intersection of parametric (hyper)surfaces . Appl. Math. Comput. 226 , 2014 , 516 – 527 .
  7. M.Düldül, On the intersection curve of three parametric hypersurfaces . Comput. Aided Geom. Des. , 27 ( 1 ), 2010 , 118 – 127 .
  8. G.Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide . Academic Press, Inc. , San Diego, CA, 2002 .
  9. H.Gluck, Higher curvatures of curves in Euclidean space . Am. Math. Mon. 73 ( 7 ), 1966 , 699 – 704 .
  10. R.Goldman, Curvature formulas for implicit curves and surfaces . Computer Aided Geometric Design 22 , 2005 , 632 – 658 .
  11. S.R.Hollasch, Four-space visualization of 4D objects. Master thesis. Arizona State University, 1991 .
  12. E.Iyigun, K.Arslan, On harmonic curvatures of curves in Lorentzian n-spaces , Commun. Fac. Sci. Univ. Ank. Series A , 54 ( 1 ), 2005 , 29 – 34 .
  13. W.Klingenberg, A Course in Differential Geometry . Springer-Verlag , New York, 1978 .
  14. Y.Y.Li, Motions of cartan curves in n-dimensional monkowski space , Modern Physics Letters A, World scientific , 28 , ( 27 ), 2013 , 1350110 – 1350123 .
  15. T.Maekawa, F. E.Wolter and N.M.Patrikalakis, Umbilics and lines of curvature for shape interrogation , Computer Aided Geometric Design , 13 , 1996 , 133 – 161 .
  16. N.M.Patrikalakis, T.Maekawa, Shape Interrogation for Computer Aided Design and Manu­facturing . Springer-Verlag , Berlin, Heidelberg, New York. 2002 .
  17. M.A.Soliman, A.Abdel, N.H.Hassan, S. A. N.Badr, Intersection curves of implicit and parametric surfaces in ℝ , Applied Mathematics 2 ( 8 ), 2011 , 1019 – 1026 .
  18. D.J.Struik, Lectures on Classical Differential Geometry . Addison-Wesley , Reading, MA, 1950 .
  19. M.Turgut, J.Luis, L.Bonilla and S.Ylmaz, On Frenet-Serret Invariants of Non-Null Curves in Lorentzian Space , International Journal of Mathematical, Computational, Physical, Elec­trical and Computer Engineering , 3 ( 7 ), 2009 , 502 – 504 .
  20. P.S.Wesson, The status of modern five dimensional gravity, arxiv.org/pdf/1412.6136. Eu­clidean 4-space , Journal of Computational and Applied Mathematics , 288 , 2015 , 81 – 98 .
  21. T.J.Willmore, An Introduction to Differential Geometry , Clarendon Press , Oxford, 1959 .
  22. Y.Xiuzi, T.Maekawa, Differential geometry of intersection curves of two surfaces , Computer Aided Geometric Design 16 1999 , 767 – 788 .
  23. S.Ylmaz, M.Turgut, A Method to Calculate Frenet Apparatus of the Curves in Euclidean-5 Space , International Journal of Computational and Mathematical Sciences , 2008 , 2 – 2 .
  24. S.Ylmaz, M.Turgut, Determination of Frenet Apparatus of Partially Null and Pseudo Null Curves in Minkowski Space-time , Int. J. Contemp. Math. Sciences , 3 ( 27 ), 2008 , 1337 – 1341 .
Views: 55Downloads: 74Citations: 1