• 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.

• 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.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.247-259
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• 2018

Factorable surfaces, i.e. graphs associated with the product of two functions of one variable, constitute a wide class of surfaces in differential geometry. Such surfaces in the pseudo-Galilean space with zero Gaussian and mean curvature were obtained in [2]. In this study, we provide new results relating to the factorable surfaces with non-zero constant Gaussian and mean curvature.

• Honam Mathematical Journal
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• v.38 no.3
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• pp.593-611
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• 2016

This paper is concerned with the classifications of quadric surfaces of first and second kinds in Euclidean 3-space satisfying the Jacobi condition with respect to their curvatures, the Gaussian curvature K, the mean curvature H, second mean curvature $H_{II}$ and second Gaussian curvature $K_{II}$. Also, we study the zero and non-zero constant curvatures of these surfaces. Furthermore, we investigated the (A, B)-Weingarten, (A, B)-linear Weingarten as well as some special ($C^2$, K) and $(C^2,\;K{\sqrt{K}})$-nonlinear Weingarten quadric surfaces in $E^3$, where $A{\neq}B$, A, $B{\in}{K,H,H_{II},K_{II}}$ and $C{\in}{H,H_{II},K_{II}}$. Finally, some important new lemmas are presented.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.17 no.2
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• pp.187-195
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• 1992

This paper propose on the determination of threshold values for extracting shape information of the objects. First, surface curvatures such as mean curvature and gaussian curvature is calculated from given range data. And then local surface regions are classified into the one of 8 primitives by using the sign of mean curvature H and gaussian curvature K. Also from the statistical viewpoint. the range of the zero of H and K in the range is obtained through the analysis of the relation between mean curvature and gaussian curvature. Finally, the effectiveness of the proposed mithod in this paper is demonstrated by comparing with a case, where the zero threshold is arbitrarily obtained.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.273-293
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• 2016

In this paper, we obtain that every biharmonic non-degenerate hypersurfaces in semi-Euclidean space $E^5_s$ with constant scalar curvature of diagonal shape operator has zero mean curvature.

• Journal of Korea Multimedia Society
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• v.20 no.12
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• pp.1951-1959
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• 2017

Due to the fact that 3D printing is applied to many areas of life, 3D printing models are often used illegally without any permission from the original providers. This paper presents a novel watermarking algorithm for the copyright protection and ownership identification for 3D printing based on the radius curvature of 3D triangle. 3D triangles are extracted and classified into groups based on radius curvature by the clustering algorithm, and then the mean radius curvature of each group will be computed for watermark embedding. The watermark data is embedded to the groups of 3D triangle by changing the mean radius curvature of each group. In each group, we select a 3D triangle which has the nearest radius curvature with the changed mean radius curvature. Finally, we change the vertices of the selected facet according to the changed radius curvature has been embedded watermark. In experiments, the distance error between the original 3D printing model and the watermarked 3D printing model is approximate zero, and the Bit Error Rate is also very low. From experimental results, we verify that the proposed algorithm is invisible and robustness with geometric attacks rotation, scaling and translation.

• Bulletin of the Korean Mathematical Society
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• v.14 no.1
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• pp.27-32
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• 1977

• Honam Mathematical Journal
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• v.46 no.2
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• pp.279-290
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• 2024

In this study, tubular surfaces were introduced according to the modified orthogonal frame defined at the points where the torsion is different from zero in the 3-dimensional Euclidean space. First, the relations between the Frenet frame and the modified orthogonal frame with torsion are given. Then, the singularity, Gaussian curvature, mean curvature and basic forms of the tubular surface given according to the modified orthogonal frame with torsion were calculated. In addition, the conditions for the parameter curves of the tubular surface to be geodesic, asymptotic and line of curvature were examined. Finally, tubular surface examples based on both the Frenet frame and the modified orthogonal frame with torsion were given to support the study.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1235-1242
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• 2013

We give area and height estimates for cmc-graphs over a bounded planar $C^{2,{\alpha}}$ domain ${\Omega}{\subset}\mathbb{R}^3$. For a constant H satisfying $H^2{\mid}{\Omega}{\mid}{\leq}9{\pi}/16$, we show that the height $h$ of H-graphs over ${\Omega}$ with vanishing boundary satisfies ${\mid}h{\mid}$ < $(\tilde{r}/2{\pi})H{\mid}{\Omega}{\mid}$, where $\tilde{r}$ is the middle zero of $(x-1)(H^2{\mid}{\Omega}{\mid}(x+2)^2-9{\pi}(x-1))$. We use this height estimate to prove the following existence result for cmc H-graphs: for a constant H satisfying $H^2{\mid}{\Omega}{\mid}$ < $(\sqrt{297}-13){\pi}/8$, there exists an H-graph with vanishing boundary.