A new approach to special curved surface families according to modified orthogonal frame

Gülnur Şaffak Atalay; Gülnur Şaffak Atalay

doi:10.3934/math.20241004

AIMS Mathematics

2024, Volume 9, Issue 8: 20662-20676. doi: 10.3934/math.20241004

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Research article

A new approach to special curved surface families according to modified orthogonal frame

Gülnur Şaffak Atalay ^,

Department of Mathematics and Science Education, Faculty of Education, Ondokuz Mayıs University, Samsun 55270, Turkey

Received: 04 April 2024 Revised: 10 June 2024 Accepted: 14 June 2024 Published: 25 June 2024
MSC : 53A04, 53A05

The main purpose of this paper was to investigate the problem of finding the surface family with respect to two different types of modified orthogonal frames defined for curves with curvature and torsion different from zero, respectively. For this purpose, conditions were given for the parametric curve with the modified orthogonal frame in three-dimensional Euclidean space to be a geodesic, asymptotic or line of curvature on the surface, respectively. It has been shown that a member of the surface family with the same special curve such as geodesic, asymptotic, or line of curvature can be obtained by choosing different deviation functions in the parametric writing of the surface to satisfy the conditions. Finally, several examples were given to support the study.
- Modified orthogonal frame,
- geodesic curve,
- asymptotic curve,
- line of curvature,
- surface family
Citation: Gülnur Şaffak Atalay. A new approach to special curved surface families according to modified orthogonal frame[J]. AIMS Mathematics, 2024, 9(8): 20662-20676. doi: 10.3934/math.20241004

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Abstract

The main purpose of this paper was to investigate the problem of finding the surface family with respect to two different types of modified orthogonal frames defined for curves with curvature and torsion different from zero, respectively. For this purpose, conditions were given for the parametric curve with the modified orthogonal frame in three-dimensional Euclidean space to be a geodesic, asymptotic or line of curvature on the surface, respectively. It has been shown that a member of the surface family with the same special curve such as geodesic, asymptotic, or line of curvature can be obtained by choosing different deviation functions in the parametric writing of the surface to satisfy the conditions. Finally, several examples were given to support the study.

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