Tangential intersection curves of two surfaces in the three-dimensional Lorentz-Minkowski space
DOI:
https://doi.org/10.17268/sel.mat.2025.01.09Keywords:
Euler Rodrigues formula, Tangential Intersection, Lorentz Minkowski space, Surface-surface intersectionAbstract
We present algorithms for computing the differential geometry properties of tangential intersection curves of two surfaces in the three-dimensional Lorentz-Minkowski space E31 . We compute the tangent vector of tangential intersection curves of two surfaces parametric, where the surfaces can be: spacelike, timelike, or lightlike. The first method computed the tangent vector using the equality of the projection of the second derivative vector onto the normal vector and second method computes the tangent vector by applying a rotation to a vector projected onto the tangent space, where the axis of rotation is the normal vector of the surface. In Minkowski space, there are three types of rotations, since the normal vectors can be: spacelike, lightlike, or timelike.
References
Bükcü Bahaddin. On the rotation matrices in the semi-euclidean space. Communications, Faculty Of Science, University of Ankara Series A1 Mathematics and Statistics, 2006, page 007–013.
Kula L, Yayli Y. Dual split quaternions and screw motion in minkowski 3-space. Iranian Journal of Science and Technology (Sciences). 2006; 30(3).
Ödemir M, Ergin AA. Rotations with unit timelike quaternions in minkowski 3-space. J. of Geometry and Physics. 2006; 56(2):322–336.
Özkaldi S, Gündogan H. Cayley Formula, Euler Parameters and Rotations in 3-Dimensional Lorentzian Space. Advances in Applied Clifford Algebras, February 2009; 20(2):367–377.
Kahvecí D, Yayli Y, Gök I. The geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space. International Journal of Geometric Methods in Modern Physics. October 2016; 13(10).
Düldül Bahar U, Düldül M. Can we find Willmore-like method for the tangential intersection problems?. J. of Computational and Applied Mathematics. August 2016; 302:301–311.
Lone M S, Shahid MH, Sharma DS. A new approach towards transversal intersection curves of two surfaces in R3. Geometry, Imaging and Computing. 2016; 3(3–4):81–99.
Lone MS, Lone MA, Shahid MH. A new approach towards geodesic curvature and geodesic torsion of transversal intersection in R3. Facta Universitatis, Series: Mathematics and Informatics. 2016; 31(3):741–749.
Do Carmo MP. Differential geometry of curves and surfaces. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.
Spivak M. A comprehensive introduction to differential geometry. Vol. III. second edition.Wilmington, Del.: Publish or Perish, Inc.; 1979.
Stoker JJ. Differential geometry. Wiley Classics Library. New York, John Wiley & Sons, Inc.; 1989.
Struik DJ. Lectures on classical differential geometry. second edition. New York: Dover Publications, Inc.; 1988.
Willmore TJ. An introduction to differential geometry. Oxford: Clarendon Press; 1959.
Faux ID, Pratt MJ. Computational geometry for design and manufacture. New York-Chichester-Brisbane: Ellis Horwood Ltd., Chichester, distributed by Halsted Press [John Wiley & Sons]; 1979.
Farin G. Curves and surfaces for computer-aided geometric design. Computer Science and Scientific Computing. Academic Press; 1997.
Hoschek J, Lasser D. Fundamentals of computer aided geometric design. A K Peters, Ltd., Wellesley, MA; 1993.
Patrikalakis NM, Maekawa T. Shape interrogation for computer aided design and manufacturing. Berlin, Springer-Verlag; 2010. Paperback reprint of the 2002 edition.
Couto IT, Lymberopoulos A. Introduction to Lorentz Geometry. London: CRC Press; 2020.
Kobayashi O. Maximal surfaces in the 3-dimensional Minkowski space L3. Tokyo J. of Mathematics. 1983; 6(2) 297-309.
Kühnel W. Differential Geometry: curves-surfaces-manifolds. 4th Edition, Vieweg, Wiesbaden; 2008.
O'Neill B. Semi-Riemannian geometry. Vol. 103 of Pure and Applied Mathematics, New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]; with applications to relativity; 1983.
Walrave J. Curves and surfaces in Minkowski space[Thesis (Ph.D.)]. Belgium: Katholieke Universiteit Leuven. ProQuest LLC, Ann Arbor, MI; 1995.
Kazaz M. Ugurlu HH, Önder M, Kahraman T. Mannheim partner d-curves in Minkowski 3-space E31. arXiv:1003.2043v3[math.DG], 2010.
Kazaz M, Ugurlu HH, Onder M, Oral S. Bertrand partner d-curves in Minkowski 3-space E31. arXiv:1003.2048v3 [math.DG], 2010.
López R. Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int. Electronic J. of Geometry. 2014; 7(1):44–107.
López R. Differential geometry of curves and surfaces in lorentz-minkowski space. arXiv:0810.3351v2 [math.DG]; 2014.
Özdemir M, Ergin AA. Spacelike Darboux curves in Minkowski 3-space. Differential Geometry-Dynamical Systems. 2007; 9:131–137.
Ugurlu H, Kocayigit H. The frenet and darboux instantaneous rotation vectors of curves on a time-like surfaces. Mathematical & Computational Aplications. 1999; 1(2):133–141.
Yilmaz S, Turgut M. On the differential geometry of curves in Minkowski space-time I. Int. J. Contemporaneo Mathematics Sciences. 2008; 3(27):1343–1349.
Yılmaz ES, Turgut M. On the differential geometry of curves in Minkowski space-time II. World Academy of Science, Engineering and Technology. 2009; 3:11–27.
Hartmann E. G2 interpolation and blending of curves. The Visual Computer. 1996; 12(4):181–192.
Ye X, Maekawa T. Differential geometry of intersection curves of two surfaces. Computer Aided Geometric Design. 1999; 16(8):767–788.
Goldman R. Curvature formulas for implicit curves and surfaces. Computer Aided Geometric Design. 2005; 22(7):632–658.
Aléssio O. Differential geometry of intersection curves for two implicit surfaces. TEMA Tend. Mat. Apl. Comput. 2006; 7(2):169-178.
Aléssio O. Differential geometry of intersection curves in R4 of three implicit surfaces. Computer Aided Geometric Design. 2009; 26(4):455-471.
Düldül BU. Caliskan M. On the intersection curve of three parametric hypersurfaces. Acta Univ. Apulensis Math. Inform. 2010; 24:161–172.
Düldül M. On the intersection curve of three parametric hypersurfaces. Computer Aided Geometric Design. 2010; 27(1):118–127.
Abdel-All NH, Badr SA-N, Soliman MA, Hassan SA. Intersection curves of hypersurfaces in R4. Computer Aided Geometric Design. 2012; 29(2):99-–108.
Düldül BU, Düldül M. The extension of Willmore’s method into 4-space. Mathematical Communications. 2012; 17(2):423-431.
Aléssio O. Formulas for second curvature, third curvature, normal curvature, first geodesic curvature and first geodesic torsion of implicit curve in n-dimensions. Computer Aided Geometric Design. 2012; 29(3–4):189–201.
Düldül M, Akbaba O . Willmore-like methods for the intersection of parametric hypersurfaces. Applied Mathematics and Computation. 2014; 226:516–527.
Calıskan M, Düldül B. On the geodesic torsion of a tangential intersection curve of two surfaces in R3. Acta Mathematica Universitatis Comenianae. 2017; 82(2):177-–189.
Abdel-All NH, Badr, SA-N, Soliman MA, Hassan SA. Intersection curves of two implicit surfaces in R3. J. of Mathematical and Computational Science. 2012; 2:152-–171.
Aléssio O, Düldül M, Düldül BU, Badr SA-N, Abdel-All NH. Differential geometry of non-transversal intersection curves of three parametric hypersurfaces in Euclidean 4-space. Comput. Aided Geom. Design. 2014; 31(9):712-727.
Badr SA-N, Abdel-All NH, Alessio O, Düldül M, Düldül BU. Non-transversal intersection curves of hypersurfaces in Euclidean 4-space. J. of Computational and Applied Mathematics. 2015; 288:81-–98.
Aléssio O, Düldül M, Düldül BU, Abdel-All NH, Badr SA-N. Differential geometry of non-transversal intersection curves of three implicit hypersurfaces in euclidean 4-space. J. of Computational and Applied Mathematics. 2016; 308:20-38.
Aléssio O, Guadalupe IV. Determination of a transversal intersection curve of two spacelike surfaces in Lorentz-Minkowski 3-space L3. Hadronic Journal. 2007; 30(3):315-341.
Düldül BU, Calıskan M. Spacelike intersection curve of three spacelike hypersurfaces in E41 . Annales. Universitatis Mariae Curie-Sklodowska. Sectio A. Mathematica. 2013; 67(1):23-33.
As E, Sarioglugil A. On integral invariants of ruled surface generated by the darboux frame of the transversal intersection timelike curve of two timelike surfaces in lorentz-minkowski 3-space. African Journal of Mathematics and Computer Science Research. 2014; 7(2):31-40.
Sanh Z, Yayh Y. Non-null intersection curves of time-like surfaces in Lorentz-Minkowski 3-space. International Journal of Engineering and Applied Sciences. 2014; 1(3):23-26.
Karaametoglu S, Aydemir I. On the transversal intersection curve of spacelike and timelike surfaces in minkowski 3-space. J. of Science and Art. 2016; 16(4):345-352.
Aléssio O, Badr SA, Hassan SA, Rodrigues LA, Silva FN, Soliman MA. Transversal intersection curves of two surfaces in minkowski 3-space. Selecciones Matem´aticas. 2018; 4(2):137-153.
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.
