Surfaces family with common null asymptotic

Abstract

We analyzed the problem of finding a surfaces family through an asymptotic curve with Cartan frame. We obtain the parametric representation for surfaces family whose members have the same as an asymptotic curve. By using the Cartan frame of the given null curve, we present the surface as a linear combination of this frame and analysed the necessary and sufficient condition for that curve to satisfy the asymptotic requirement. We illustrate the method by giving some examples.

An approach for designing a surface pencil through a given geodesic curve

“…Also, surfaces with common geodesic in Minkowski 3-space have been the subject of many studies. In [31], Kasap and Aky¬ld¬z de…ned surfaces with a common geodesic in Minkowski 3-space and gave the su¢ cient conditions on marching-scale functions so that the given curve is a common geodesic on that surfaces.Şa¤ak and Kasap [32,33] studied family of surfaces with a common null geodesic and null asymptotic. Bayram et al [34] studied parametric surfaces which possess a given curve as a common asymptotic.…”

“…In addition, they speci…cally stated their work for ruled surfaces and gave the conditions to be developable within them. Also, recently Atalay [36,37] studied surface family possessing a Mannheim pair of a given curve as an asymptotic and geodesic curve. She, express necessary and su¢ cient conditions for that curve with above property and present natural results for such ruled surfaces.…”

Surface Family With Common Line of Curvature in 3-Dimensional Galilean Space

“…This method can be used for obtaining minimal cost of material while building surfaces. The surface family notion has been studied by many researchers [1,2,9,10].…”

Surface pencil with a common adjoint curve

“…Beside the Euclidean space, there exist researches about surface pencils in other spaces. Atalay and Kasap [12] expressed necessary and sufficient conditions for surfaces to have the same curve as a null asymptotic curve in Minkowski 3-space. Yüzbaşı [7] obtained constraints for family of surfaces through a given asymptotic curve in Galilean 3-space.…”