On Frenet Apparatus of Curves in Rn
DOI:
https://doi.org/10.14244/lajm.v1i01.3Keywords:
curve, Frenet apparatus, curvature, hypersurfacesAbstract
In this paper, we present new explicit and nonrecursive formulas for the curvatures and the frame of Frenet of a regular curve with an arbitrary parameter in the Euclidean space Rn, n > 2, expressed only in terms of its derivatives.
References
Barbosa L. & do Carmo M. Helicoids, catenoids, and minimal hypersurfaces of Rn invariant by an one-parameter group of motions. An. Acad. Brasil. Cieˆncias., 53 (1981), 403-408.
do Carmo, M. (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs.
Flanders, H. (1963). Differential Forms with Applications to the Physical Sciences. Dover Publication, New York.
Greub, W. (1975). Linear Algebra. Springer-Verlag New York Inc 1975. DOI: https://doi.org/10.1007/978-1-4684-9446-4
Gluck, H. (1966). Higher curvatures of curves in euclidean space. DOI: https://doi.org/10.2307/2313974
Kühnel, W. (2002) Differential geometry, curves-surfaces-manifolds, Translated from the (1999) German original by Bruce Hunt. Student Mathematical Library, Vol. 16, American Mathematical Society, Providence, RI, 2002.
Kuttler, K. (2012) Linear algebra: theory and applications The Saylor Foundation, 2012. [8] Lee, J. M. Manifolds and Differential Geometry. Graduate studies in mathematics. American Mathematical Society, 2009.
Lima, Elon L. (1965) Cálculo tensorial IMPA, RJ, Brasil, 2012.
O’Neill, B. (1966). Elementary Differential Geometry. Academic Press, New York. DOI: https://doi.org/10.1016/B978-1-4832-3170-9.50011-7
Steven J. Leon, Ake Björck, and Walter Gander. Gram-Schmidt orthogonalization: 100 years and more. Numerical Linear Algebra with Applications, 20(3):492–532, 2013. DOI: https://doi.org/10.1002/nla.1839
Sulanke, R. (2020). The Fundamental Theorem for Curves in the n-Dimensional Euclidean Space. Preprint.
Wolfram, S. (2019). Mathematica Software, Version 12.9.
Süha Yilmaz, S. & Turgut, M. (2008). A method to calculate Frenet apparatus of the curves in euclidean-5 space. International Scholarly and Scientific Research & Innovation, 2, No:7:483–485. Numerical Linear Algebra with Applications, 73:4:699– 704, DOI: 10.1080/00029890.1966.11970818. DOI: https://doi.org/10.1080/00029890.1966.11970818
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Isnaldo Barbosa, José Adonai Pereira Seixas
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors may enter into separate contractual agreements for the non-exclusive distribution of the journal's published version of the work (for example, posting it in an institutional repository or publishing it in a book), with an acknowledgment of its initial publication in this journal.
