In the present paper, we handle the problem of finding a hypersurface family from a given asymptotic curve in R 4. 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.

In this paper, we analyze the problem of constructing a surface pencil from a
given spacelike (timelike) line of curvature. By using the Frenet frame of the
given curve in Minkowski 3-space, we express the surface pencil as a linear
combination of this frame and derive the necessary and sufficient conditions
for the coefficients to satisfy the line of curvature requirement. To
illustrate the method some examples showing members of the surface pencil with
their line of curvature are given.Comment: 23 pages, 6 figure

Let [Formula: see text] be a parameter and an asymptotic curve on a surface [Formula: see text] We obtain conditions for offsets [Formula: see text] of [Formula: see text] such that the image [Formula: see text] of the curve [Formula: see text] is a common asymptotic on each offset. We illustrate the method with an example.

In the present paper, we construct surfaces possessing an adjoint curve of a given space curve as an asymptotic curve, geodesic or line of curvature. We obtain conditions for ruled surfaces and developable ones. Finally, we present illustrative examples to show the validity of the present method.

We construct a surface family possessing an involute of a given curve as an asymptotic curve. We express necessary and sufficient conditions for that curve with the above property. We also present natural results for such ruled surfaces. Finally, we illustrate the method with some examples, e.g. circles and helices as given curves.

There is a unique curve called striction curve that is not present on a surface except a ruled surface. This curve is defined as the shortest distance with the help of a common perpendicular line between two adjacent rulings. In this work, we present Lorentz forces and magnetic striction curves produced by the geodesic Frenet frame
{}ē,0.1emtruet¯,0.1emtrueg¯$$ \left\{\bar{e},\overline{t},\overline{g}\right\} $$ on the striction curve of the ruled surface defined by the spherical indicatrix curve in a magnetic field. We calculate magnetic vector fields of magnetic striction curves for
truee‾,0.1emtruet¯$$ \overline{e},\overline{t} $$, and
trueg¯$$ \overline{g} $$. Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic striction curves. We obtain developability conditions for these surfaces according to their curvature functions. Finally, we give some examples about magnetic flux surfaces.

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