Extended Darboux frame curvatures of Frenet curves lying on parametric 3-surfaces

Bahar Uyar Düldül, Mustafa Düldül


In this paper, we consider a Frenet curve lying on a parametric hypersurface in 4-dimensional Euclidean space and obtain the expressions of its curvatures with respect to the hypersurface


extended Darboux frame field; hypersurface; curvatures

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do Carmo, M.P.: Differential Geometry of curves and surfaces. Prentice Hall, Englewood Cliffs, NJ, (1976)

D"uld"ul, M., Uyar D"uld"ul, B., Kuruou glu, N., "Ozdamar, E.: The extension of Darboux frame into Euclidean 4-space and its invariants. Turk. J. Math. doi: 10.3906/mat-1604-56, (in press)

Farouki, R.T.: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Geometry and Computing (Book 1), Springer, (2008)

Gluck, H.: Higher curvatures of curves in Euclidean space. Amer. Math. Monthly {bf 73}, 699-704, (1966)

Hollasch, S.R.: Four-space visualization of 4D objects. Master thesis, Arizona State University, (1991)

O'Neill, B.: Elementary Differential Geometry. Academic Press, (1966)

Patrikalakis, N.M., Maekawa, T.: Shape Interrogation for Computer Aided Design and Manufacturing. Springer, (2002)

Spivak, M. A.: Comprehensive Introduction to Differential Geometry. Publish or Perish, vol. 3, 3rd Edition, Houston, Texas, (1999)

Struik, D.J.: Lectures on classical differential geometry. Addison-Wesley, Reading, MA, (1950)

Williams, M.Z., Stein, F.M.: A triple product of vectors in four-space. Math. Mag. {bf 37}, 230-235, (1964)


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