The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space

Yanlin Li; Kemal Eren; Kebire Hilal Ayvacı; Soley Ersoy; Yanlin Li; Kemal Eren; Kebire Hilal Ayvacı; Soley Ersoy

doi:10.3934/math.2023115

AIMS Mathematics

2023, Volume 8, Issue 1: 2226-2239. doi: 10.3934/math.2023115

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The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space

1.
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
2.
Department of Mathematics, Faculty of Arts and Sciences, Sakarya University, Sakarya 54050, Turkey
3.
Department of Mathematics, Faculty of Arts and Sciences, Ordu University, Ordu, Turkey

Received: 03 September 2022 Revised: 10 October 2022 Accepted: 21 October 2022 Published: 31 October 2022
MSC : 53A04, 53A05, 58K05

In this study, the ruled developable surfaces with pointwise 1-type Gauss map of Frenet-type framed base (Ftfb) curve are introduced in Euclidean 3-space. The tangent developable surfaces, focal developable surfaces, and rectifying developable surfaces with singular points are considered. Then the conditions for the Gauss map of these surfaces to be pointwise 1-type are obtained separately. In order to form a basis for the study, first, the basic concepts related to the Ftfb curve and the Gauss map of a surface are recalled. Later, the necessary and sufficient conditions are found for these surfaces to be of the pointwise 1-type of the Gauss map. Finally, examples for each type of these surfaces are given, and their graphics are illustrated.
- Gauss maps,
- tangent developable surfaces,
- focal developable surfaces,
- rectifying developable surfaces,
- Frenet type frame base curves,
- singular points
Citation: Yanlin Li, Kemal Eren, Kebire Hilal Ayvacı, Soley Ersoy. The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space[J]. AIMS Mathematics, 2023, 8(1): 2226-2239. doi: 10.3934/math.2023115

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Abstract

In this study, the ruled developable surfaces with pointwise 1-type Gauss map of Frenet-type framed base (Ftfb) curve are introduced in Euclidean 3-space. The tangent developable surfaces, focal developable surfaces, and rectifying developable surfaces with singular points are considered. Then the conditions for the Gauss map of these surfaces to be pointwise 1-type are obtained separately. In order to form a basis for the study, first, the basic concepts related to the Ftfb curve and the Gauss map of a surface are recalled. Later, the necessary and sufficient conditions are found for these surfaces to be of the pointwise 1-type of the Gauss map. Finally, examples for each type of these surfaces are given, and their graphics are illustrated.

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