• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.509-531
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• 2022

In this paper we define a new family of ruled surfaces using an othonormal Sannia frame defined on a base consisting of the striction curve of the tangent, the principal normal, the binormal and the Darboux ruled surface. We examine characterizations of these surfaces by first and second fundamental forms, and mean and Gaussian curvatures. Based on these characterizations, we provide conditions under which these ruled surfaces are developable and minimal. Finally, we present some examples and pictures of each of the corresponding ruled surfaces.

• Journal of the Korea Computer Graphics Society
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• v.4 no.2
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• pp.69-75
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• 1998

Ruled surfaces are useful concept for surface design because they are defined by the one-parameter family of lines. In this paper, we prove that the offsets of a developable surface (a special class of ruled surfaces) are developable surfaces. Moreover, we prove that the offsets of a non-developable ruled surface cannot be ruled surfaces.

• Journal of the Korean Mathematical Society
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• v.55 no.4
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• pp.987-1004
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• 2018

In this paper, we get several centroaffine invariant properties for a ruled surface in ${\mathbb{R}}^4$ with centroaffine theories of codimension two. Then by solving certain partial differential equations and studying a centroaffine surface with some centroaffine invariant properties in ${\mathbb{R}}^4$, we obtain such a surface is centroaffinely equivalent to a ruled surface or one of the flat centered cyclic surfaces. Furthermore, some centroaffine invariant properties for centered cyclic surfaces are considered.

• Honam Mathematical Journal
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• v.44 no.4
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• pp.594-617
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• 2022

In this study, firstly Smarandache ruled surfaces whose base curves are Smarandache curves derived from Frenet vectors of the curve, and whose direction vectors are unit vectors plotting Smarandache curves, are created. Then, the Gaussian and mean curvatures of the obtained ruled surfaces are calculated separately, and the conditions to be developable or minimal for the surfaces are given. Finally, the examples are given for each surface and the graphs of these surfaces are drawn.

• Kyungpook Mathematical Journal
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• v.47 no.4
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• pp.579-593
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• 2007

In this paper, we study some properties of ruled surfaces in a three-dimensional Lorentz-Minkowski space related to their Gaussian curvature, the second Gaussian curvature and the mean curvature. Furthermore, we examine the ruled surfaces in a three-dimensional Lorentz-Minkowski space satisfying the Jacobi condition formed with those curvatures, which are called the II-W and the II-G ruled surfaces and give a classification of such ruled surfaces in a three-dimensional Lorentz-Minkowski space.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1661-1668
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• 2015

We study the Gauss map G of ruled surfaces in the 3-dimensional Euclidean space $\mathbb{E}^3$ with respect to the so called Cheng-Yau operator ${\Box}$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only ruled surfaces with Gauss map G satisfying ${\Box}G=AG$ for some $3{\times}3$ matrix A are the flat ones. Furthermore, we show that the only ruled surfaces with Gauss map G satisfying ${\Box}G=AG$ for some nonzero $3{\times}3$ matrix A are the cylindrical surfaces.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.177-183
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• 2009

In this article, we study ruled surfaces in a Lorentzian space-time, which have finite type immersion. We give a condition for bi-finite type surfaces to be of finite type.

• Honam Mathematical Journal
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• v.41 no.1
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• pp.185-195
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• 2019

In this work, we study the conditions between null geodesic curves and timelike ruled surfaces in dual Lorentzian space. For this study, we establish a system of differential equations characterizing timelike ruled surfaces in dual Lorentzian space by using the invariant quantities of null geodesic curves on the given ruled surfaces. We obtain the solutions of these systems for special cases. Regarding to these special solutions, we give some results of the relations between null geodesic curves and timelike ruled surfaces in dual Lorentzian space.

• Honam Mathematical Journal
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• v.30 no.4
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• pp.637-644
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• 2008

In this paper, we study some properties about the parallel surfaces of ruled surfaces in a Euclidean 3-space. Furthermore, we classify the parallel surfaces of ruled surfaces in a Euclidean 3-space satisfying a linear type and a quadric type with respect to the Gaussian curvature and the mean curvature.

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.753-761
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• 2001

We introduce the notion of Gauss map of pointwise 1-type on ruled surfaces in the Euclidean 3-space for which vector valued functions is neither trivial nor it extends or coincides with the usual notion of 1-type, in general. We characterize the minimal helicoid in terms of it and give a complete classification of the ruled surfaces with pointwise 1-type Gauss map.