• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.1-20
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• 2021

We examine the theory of surfaces in the isotropic three-space, with emphases on the surfaces related to the zero mean curvature.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.159-175
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• 2017

We solve the $Bj{\ddot{o}}rling$ problem for constant mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space. That is, we show that for any regular, analytic (and spacelike in the case of de Sitter three-space) curve ${\gamma}$ and an analytic (timelike in the case of de Sitter three-space) unit vector field N along and orthogonal to ${\gamma}$, there exists a unique (spacelike in the case of de Sitter three-space) surface of constant mean curvature 1 which contains ${\gamma}$ and the unit normal of which on ${\gamma}$ is N. Some of the consequences are the planar reflection principles, and a classification of rotationally invariant CMC 1 surfaces.

• Honam Mathematical Journal
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• v.40 no.4
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• pp.701-718
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• 2018

In this paper, we will study the constant mean curvature (CMC) surfaces foliated by ellipses in three dimensional Euclidean space $E^3$. We prove that: (1): Surfaces foliated by ellipses are CMC surfaces if and only if it is a part of generalized cylinder. (2): All surfaces foliated by ellipses are not minimal surfaces. (3): CMC surfaces foliated by ellipses are developable surfaces. (4): CMC surfaces foliated by ellipses are translation surfaces generated by a straight line and plane curve.

• Kyungpook Mathematical Journal
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• v.58 no.3
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• pp.547-557
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• 2018

We establish some characterizations of isoparametric surfaces in the three-dimensional Euclidean space, which are associated with the Laplacian operator defined by the so-called II-metric on surfaces with non-degenerate second fundamental form and the elliptic linear Weingarten metric on surfaces in the three-dimensional Euclidean space. We also study a Ricci soliton associated with the elliptic linear Weingarten metric.

• Bulletin of the Korean Mathematical Society
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• v.61 no.2
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• pp.451-467
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• 2024

We solve the Björling problem for zero mean curvature surfaces in the three-dimensional light cone. As an application, we construct and classify all rotational zero mean curvature surfaces.

• 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.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1235-1255
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• 2018

We provide definitions for the helicoidal Killing field and the helicoid in arbitrary three-manifolds, and investigate helicoids and ruled minimal surfaces in homogeneous three-manifolds, mainly in $SL_2{\mathbb{R}}$ and Sol(3). In so doing we finish our classification of ruled minimal surfaces in homogeneous three-manifolds with the isometry group of dimension 4.

• Korean Journal of Computational Design and Engineering
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• v.10 no.1
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• pp.27-39
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• 2005

Implicit surfaces are geometric shapes which are defined by implicit functions and exist in three-dimensional space. Recently, implicit surfaces have received much attention in solid modeling applications because they are easy to represent the location of points and to use boolean operations. However, it is difficult to chart points on implicit surfaces for rendering. As efficient rendering method of implicit surfaces, the original Adaptively Sampled Distance Fields (ADFs) $method^{[1]}$ is to use sampled distance fields which subdivide the three dimensional space of implicit surfaces into many cells with high sampling rates in regions where the distance field contains fine detail and low sampling rates where the field varies smoothly. In this paper, in order to maintain the sharp features efficiently with small number of cells, an extended ADFs method is proposed, applying the Dual/Primal mesh optimization $method^{[2]}$ to the original ADFs method. The Dual/Primal mesh optimization method maintains sharp features, moving the vertices to tangent plane of implicit surfaces and reconstructing the vertices by applying a curvature-weighted factor. The proposed extended ADFs method is applied to several examples of implicit surfaces to evaluate the efficiency of the rendering performance.

• Journal of Biosystems Engineering
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• v.17 no.1
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• pp.37-44
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• 1992

This study was intended to develop the design program of the working surface of moldboard-plow by use of the computer-aided design. The mathematical model of the working surfaces of moldboard-plows by use of computer graphics was developed and plotted in two dimension on three major planes. The surfaces of moldboard-plows were represented with "B-spline surface fitting" by selecting the twenty-five three-dimensional data that could well describe the working surface of moldboard-plow. The shape of moldboard-plow on three major planes was drawn for varied design parameters. The representation of the mathematical model for the working surfaces of various types of moldboard-plows was manipulated by translation, rotation and scaling about arbitrary axes in space. By using three-dimensional graphics techique to describe moldboard-plows, it was capable of plotting the three-dimensional shape of moldboard-plow easily and quickly in comparison with the existing design methods that were difficult to grasp the shape of moldboard-plow as a whole. The design theories of moldboard plow and three-dimensional computer graphic technique were applied to find out the improved reversible Jaenggi bottom. It was resulted in the newly developed shape of Jaenggi which may be used for improving the performance compared to existing ones.

• Korean Journal of Computational Design and Engineering
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• v.6 no.2
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• pp.125-132
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• 2001

An algorithm for generating all hexahedral meshes for three dimensional objects has been presented. This algorithm is based on the sweeping and the grafting method. In sweeping process internal nodes generating method has been modified by employing the distances between nodes on connecting surfaces and on source surfaces. In addition to the sweeping processes grafting algorithm is also modified to obtain more effective meshes by refining elements near grafting surfaces. With this method two and a half dimensional hexahedral meshes for three dimensional objects can be generated effectively. Sample meshes are constructed to demonstrate the mesh generating capability of the proposed algorithm.