Abstract
We handle the problem of finding a hypersurface family from a given asymptotic curve in R4. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. We illustrate this method by presenting some examples.
Highlights
Asymptotic curves are encountered in differential geometry frequently
A surface curve is called asymptotic if its tangent vectors always point in an asymptotic direction, that is, the direction in which the normal curvature is zero
Asymptotic curves on a surface can be seen in many differential geometry books [1,2,3,4,5]
Summary
Asymptotic curves are encountered in differential geometry frequently. A surface curve is called asymptotic if its tangent vectors always point in an asymptotic direction, that is, the direction in which the normal curvature is zero. Romero-Fuster et al [3] studied asymptotic curves on generally immersed surfaces in R5. Bayram et al [8] tackled the problem of finding a surface pencil from a given asymptotic curve. There is little literature on differential geometry of parametric surface family in R3 [8, 13,14,15,16] but not in R4. We consider the four-dimensional analogue problem of constructing a parametric representation of a surface family from a given asymptotic as in Bayram et al [8], who derived the necessary and sufficient conditions on the marching-scale functions for which the curve C is an asymptotic curve on a given surface. We find the necessary and sufficient constraints on the marching-scale functions, namely, coefficients of Frenet vectors, so that both the asymptotic and parametric requirements are met
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