Some Characterizations of Isotropic Curves In the Euclidean Space
The curves, of which the square of the distance
between the two points equal to zero, are called minimal or isotropic
curves [4]. In this work, first, necessary and sufficient conditions to
be a Pseudo Helix, which is a special case of such curves, are
presented. Thereafter, it is proven that an isotropic curve-s position
vector and pseudo curvature satisfy a vector differential equation of
fourth order. Additionally, In view of solution of mentioned
equation, position vector of pseudo helices is obtained.
[1] W. Blaschke and H. Reichard, Einfuhrung in die Differential Geometrie,
Berlin-Gottingen-Heidelberg, 1960.
[2] C. Boyer, A History of Mathematics, New York: Wiley,1968.
[3] U. Pekmen, "On Minimal Space Curves in the Sense of Bertrand
Curves", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Vol.10, pp.
3-8 ,1999.
[4] F. Semin, Differential Geometry I, Istanbul University, Science Faculty
Press, 1983.
[5] D. Struik, Lectures on Classical Differential Geometry I, America, 1961.
[6] S. Yilmaz, S. Nizamoglu and M. Turgut, "A Note on Differential
Geometry of the Curves in E4 ", Int. J. Math. Comb. Vol. 2, pp. 104-
108, 2008.
[1] W. Blaschke and H. Reichard, Einfuhrung in die Differential Geometrie,
Berlin-Gottingen-Heidelberg, 1960.
[2] C. Boyer, A History of Mathematics, New York: Wiley,1968.
[3] U. Pekmen, "On Minimal Space Curves in the Sense of Bertrand
Curves", Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Vol.10, pp.
3-8 ,1999.
[4] F. Semin, Differential Geometry I, Istanbul University, Science Faculty
Press, 1983.
[5] D. Struik, Lectures on Classical Differential Geometry I, America, 1961.
[6] S. Yilmaz, S. Nizamoglu and M. Turgut, "A Note on Differential
Geometry of the Curves in E4 ", Int. J. Math. Comb. Vol. 2, pp. 104-
108, 2008.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:50684", author = "Süha Yılmaz and Melih Turgut", title = "Some Characterizations of Isotropic Curves In the Euclidean Space", abstract = "The curves, of which the square of the distance
between the two points equal to zero, are called minimal or isotropic
curves [4]. In this work, first, necessary and sufficient conditions to
be a Pseudo Helix, which is a special case of such curves, are
presented. Thereafter, it is proven that an isotropic curve-s position
vector and pseudo curvature satisfy a vector differential equation of
fourth order. Additionally, In view of solution of mentioned
equation, position vector of pseudo helices is obtained.", keywords = "Classical Differential Geometry, Euclidean space,
Minimal Curves, Isotropic Curves, Pseudo Helix.", volume = "2", number = "7", pages = "374-3", }