• 호남수학학술지
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• 제32권1호
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• pp.53-60
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• 2010

The concept of discrete curvature is a discretization of the curvature. Many literatures introduce discrete curvatures derived from arc length of circular arcs. We propose a new concept of discrete curvature of a polygon at each vertex, which is derived from area of fan shapes. We estimate the error of the discrete curvature and compare the discrete curvature with old one.

• 호남수학학술지
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• 제37권4호
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• pp.487-504
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• 2015

For a smooth plane curve, the curvature can be characterized by the rate of change of the angle between the tangent vector and a fixed vector. In this article we prove that the curvature of a space curve can also be given by the rate of change of the locally defined angle between the tangent vector at a point and the nearby point. By using height functions, we introduce turning angle of a space curve and characterize the curvature by the rate of change of the turning angle. The main advantage of the turning angle is that it can be used to characterize the curvature of discrete curves. For this purpose, we introduce a discrete turning angle and a discrete curvature called visual curvature for space curves. We can show that the visual curvature is an approximation of curvature for smooth curves.

• Journal of applied mathematics & informatics
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• 제18권1_2호
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• pp.235-246
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• 2005

We present an interpolating, univariate subdivision scheme which preserves the discrete curvature and tangent direction at each step of subdivision. Since the polygon have a geometric information of some original(in some sense) curve as a discrete curvature, we can expect that the limit curve has the same curvature at each vertex as the control polygon. We estimate the curvature bound of odd vertices and give an error estimate for restoring a curve from sampled vertices on curves.

• 충청수학회지
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• 제34권1호
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• pp.37-53
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• 2021

In this paper we define scaled visual curvature and visual Frenet frame that can be visually accepted for discrete space curves. Scaled visual curvature is relatively simple compared to multi-scale visual curvature and easy to control the influence of noise. We adopt scaled minimizing directions of height functions on each neighborhood. Minimizing direction at a point of a curve is a direction that makes the point a local minimum. Minimizing direction can be given by a small noise around the point. To reduce this kind of influence of noise we exmine the direction whether it makes the point minimum in a neighborhood of some size. If this happens we call the direction scaled minimizing direction of C at p ∈ C in a neighborhood Br(p). Normal vector of a space curve is a second derivative of the curve but we characterize the normal vector of a curve by an integration of minimizing directions. Since integration is more robust to noise, we can find more robust definition of discrete normal vector, visual normal vector. On the other hand, the set of minimizing directions span the normal plane in the case of smooth curve. So we can find the tangent vector from minimizing directions. This lead to the definition of visual tangent vector which is orthogonal to the visual normal vector. By the cross product of visual tangent vector and visual normal vector, we can define visual binormal vector and form a Frenet frame. We examine these concepts to some discrete curve with noise and can see that the scaled visual curvature and visual Frenet frame approximate the original geometric invariants.

• 컴퓨터교육학회논문지
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• 제9권1호
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• pp.89-95
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• 2006

컴퓨터그래픽스에서 다루어지는 3차원 물체들에 대한 곡률과 같은 기하학적 특성들은 메쉬의 모양을 해석함에 있어 매우 중요한 역할을 한다. 부드러운 곡면에서 정의되는 곡률은 메쉬와 같은 이산적 형태에서는 수학적으로 정의할 수 없다. 그러므로 이러한 이산곡률을 어떻게 정의하느냐에 따라 기하학적 연산들의 결과는 많은 영향을 받는다. 본 논문에서는 기존의 곡률 계산법에서 사용하고 있는 단면곡률 계산법의 오류를 지적하고 이에 대한 해결책으로 베이지어 곡선을 이용한 포물선-기반 이산 곡률 계산법을 제시한다. 제안된 방법을 통하면 보다 뾰족한 형태의 정점과 완만한 형태의 정점을 구분할 수 있어서 메쉬 간략화와 같은 기하학적 연산에 쉽게 적용가능하다.

• 대한수학회지
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• 제56권1호
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• pp.113-125
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• 2019

In this paper, we consider the Ricci curvature of a directed graph, based on Lin-Lu-Yau's definition. We give some properties of the Ricci curvature, including conditions for a directed regular graph to be Ricci-flat. Moreover, we calculate the Ricci curvature of the cartesian product of directed graphs.

• 한국정보과학회논문지:시스템및이론
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• 제27권3호
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• pp.245-254
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• 2000

본 논문은 LOD 메쉬 생성을 위한 이산 곡률을 이용한 새로운 오차 척도를 제안한다. 메쉬의 간략화를 위한 이산 곡률은, 부드러운 곡면 추정의 과정 없이 꼭지점 중심의 표면각과 표면적, 이면각 등 의 기하학적 속성만을 이용하여 계산되는 곡률로서, 표면의 특징을 잘 표현하고 있다. 그러므로 이산 곡률에 기반한 새로운 이산 곡률 오차 척도는 원래 모델의 기하학적 형상을 최대로 유지하여 간략화 모델의 정확성을 증가 시키고, 전역 오차 척도로 사용될 수 있다. 또한, 본 논문에서는 LOD 모델을 간략화 비율이 아닌, 오차 척도를 기준으로 생성할 것을 제안한다. 왜냐 하면 LOD는 원래 모델과 각 단계의 간략화된 모델 사이의 근접도에 따라 나누어진 단계를 뜻하기 때문이다. 따라서 이산 곡률 오차 척도는 기존의 오차 척도에 비해 비교적 많은 수학적 연산이 필요하나, 각 단계의 LOD 모델이 원래 모델의 형상을 잘 유지하면서 간략화 비율이 아닌 상세도의 차이를 가지도록 효과적으로 LOD를 생성, 제어할 수 있다.

• 대한수학회보
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• 제61권2호
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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.

• 호남수학학술지
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• 제38권2호
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• pp.359-366
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• 2016

Let H be the 3-dimensional Heisenberg group, ($G=H{\times}S^1$, g) a product Riemannian manifold of Riemannian manifolds H and S with arbitrarily given left invariant Riemannian metrics respectively, and ${\Gamma}$ the discrete subgroup of G with integer entries. Then, on the Riemannian manifold ($M:=G/{\Gamma}$, ${\Pi}^*g=\bar{g}$), ${\Pi}:G{\rightarrow}G/{\Gamma}$, we evaluate the scalar curvature and the Ricci curvature.

• Structural Engineering and Mechanics
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• 제43권2호
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• pp.147-161
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• 2012

This paper presents a curvature method for analysis of beam-columns with different materials and arbitrary cross-section shapes and subjected to combined biaxial moments and axial load. Both material and geometric nonlinearities (the p-delta effect in this case) were incorporated. The proposed method considers biaxial curvatures and uniform normal strains of discrete cross-sections of beam-columns as basic unknowns, and seeks for a solution of the column deflection curve that satisfies force equilibrium conditions. A piecewise representation of the beam-column deflection curve is constructed based on the curvatures and angles of rotation of the segmented cross-sections. The resulting bending moments were evaluated based on the deformed column shape and the axial load. The moment curvature relationship and the beam-column deflection calculation are presented in matrix form and the Newton-Raphson method is employed to ensure fast and stable convergence. Comparison with results of analytic solutions and eccentric compression tests of wood beam-columns implies that this method is reliable and effective for beam-columns subjected to eccentric compression load, lateral bracings and complex boundary conditions.