• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.891-906
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• 2015

In this work we analyze a postprocessing scheme for improving the guaranteed error bound based on the equilibrated fluxes for the P1 conforming FEM. The improved error bound is shown to be asymptotically exact under suitable conditions on the triangulations and the regularity of the true solution. We also present some numerical results to illustrate the effect of the postprocessing scheme.

• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.393-406
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• 2020

We analyze a posteriori error estimator for the conforming P2 finite element on triangular meshes which is based on the solution of local Neumann problems. This error estimator extends the one for the conforming P1 finite element proposed in [4]. We prove that it is asymptotically exact for the Poisson equation when the underlying triangulations are mildly structured and the solution is smooth enough.

• The Pure and Applied Mathematics
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• v.22 no.1
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• pp.91-99
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• 2015

A normal surface is determined by how the surface under consideration meets each tetrahedron in a given triangulation. We call such a nice embedded disk, which is a component of the intersection of the surface with a tetrahedron, an elementary disk. We classify all elementary disk types in a truncated ideal triangulation.

• Kyungpook Mathematical Journal
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• v.62 no.2
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• pp.289-321
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• 2022

The work realized by the authors of [4], [5] and [6] associates a non-negative matrix with positive integers entries to each dissection of a polygon. In the particular case of triangulations, these matrices called ℬ𝒞𝒥-matrices here contain valuable information of their frieze patterns, a concept introduced by Coxeter and Conway. This paper is concerned with the algebraic manipulation and properties of these matrices which are derived from operations acting on dissections.

• Journal of Institute of Control, Robotics and Systems
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• v.17 no.11
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• pp.1159-1167
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• 2011

A novel 3D tube scanning technique is proposed. The proposed tube scanning technique is developed for a special tube inspection module which consists of four line-lasers and one camera. Using the scanning module, we can reconstruct the 360 degree shapes of the inner surfaces of a cylindrical tube. From an image frame captured by the camera, we reconstruct a partial tube model based on four laser triangulations. Then by aligning such partial models with respect to a reference tube axis, a complete 3D shape of the tube is reconstructed. The tube axis in each reconstructed frame is aligned with a 3D Euclidean transformation to the reference axis. Several experiments show that the proposed method can align multiple tube axes very accurately and reconstruct 3D shapes of a tube with very low shape distortion.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.807-820
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• 2017

An open smooth manifold is said of finite geometric rank if it admits a handlebody decomposition with a finite number of 1-handles. We prove that, if there exists a proper submanifold $W^{n+3}$ of finite geometric rank between an open 3-manifold $V^3$ and its stabilization $V^3{\times}B^n$(where $B^n$ denotes the standard n-ball), then the manifold $V^3$ has the Tucker property. This means that for any compact submanifold $C{\subset}V^3$, the fundamental group ${\pi}_1(V^3-C)$ is finitely generated. In the irreducible case this implies that $V^3$ has a well-behaved compactification.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.105-115
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• 2003

An infinite locally finite plane graph is called an LV-graph if it is 3-connected and VAP-free. If an LV-graph G contains no unbounded faces, then we say that G is a 3LV-graph. In this paper, a structure theorem for an LV-graph concerning the existence of a sequence of systems of paths exhausting the whole graph is presented. Combining this theorem with the early result of the author, we obtain a necessary and sufficient conditions for an infinite VAP-free planar graph to be a 3LV-graph as well as an LV-graph. These theorems generalize the characterization theorem of Thomassen for infinite triangulations.

• International Journal of CAD/CAM
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• v.9 no.1
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• pp.25-29
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• 2010

Let $S_n^r(\Omega)$ be the spline space of degree n and smoothness r with respect to $\Omega$ where $\Omega$ is a triangulation of a planner polygonal domain. Dimensions of $S_n^r(\Omega)$ over the so-called unconstricted triangulation were given by Farin in [J. Comput. Appl. Math. 192(2006), 320-327]. In this paper, a counter example is given to show that the condition used in the main result in Farins paper is not correct, and then an improved necessary and sufficient condition is presented.

• Proceedings of the Korean Society of Broadcast Engineers Conference
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• 2006.11a
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• pp.59-62
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• 2006

본 논문에서는 OpenGL Rendering을 이용한 모델기반 3D 다시점 영상의 객체 구현을 위한 구성과 각 모듈에 적용되는 알고리즘에 대해 중점적으로 연구하였다. 한 장의 텍스쳐 이미지와 깊이 맵(Depth Map)을 가지고 다시점 객체를 생성하기 위해, 먼저 깊이 정보의 전처리 과정을 거친다. 전처리 된 깊이 정보는 OpenGL상에서의 일정 간격의 꼭지점(Vertex) 정보로 샘플링 된다. 샘플링 된 꼭지점 정보는 깊이 정보를 z값으로 가지는 3차원 공간 좌표상의 점이다. 이 꼭지점 정보를 기반으로 텍스쳐 맵핑 (texture mapping)을 위한 폴리곤(polygon)을 구성하기 위해 딜루이니 삼각화(Delaunay Triangulations) 알고리즘이 적용되었다. 이렇게 구성된 폴리곤 위에 텍스쳐 이미지를 맵핑하여 OpenGL의 좌표 연산을 통해 시점을 자유롭게 조정할 수 있는 객체를 만들었다. 제한된 하나의 이미지와 깊이 정보만을 가지고 좀 더 넓은 범위의 시점을 가지는 다시점 객체를 생성하기 위해, 새로운 꼭지점을 생성하여 폴리곤을 확장시켜 기존보다 더 넓은 시점을 확보할 수 있었다. 또한 렌더링된 모델의 경계 영역 부분의 깊이정보 평활화를 통해 시각적인 개선을 이룰 수 있었다.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.41 no.4
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• pp.123-130
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• 2018

Triangulation is one of the fundamental problems in computational geometry and computer graphics community, and it has huge application areas such as 3D printing, computer-aided engineering, surface reconstruction, surface visualization, and so on. The Delaunay refinement algorithm is a well-known method to generate quality triangular meshes when point cloud and/or constrained edges are given in two- or three-dimensional space. In this paper, we propose a simple but efficient algorithm to triangulate Voronoi surfaces of Voronoi diagram of spheres in 3-dimensional Euclidean space. The proposed algorithm is based on the Ruppert's Delaunay refinement algorithm, and we modified the algorithm to be applied to the triangulation of Voronoi surfaces in two ways. First, a new method to deciding the location of a newly added vertex on the surface in 3-dimensional space is proposed. Second, a new efficient but effective way of estimating approximation error between Voronoi surface and triangulation. Because the proposed algorithm generates a triangular mesh for Voronoi surfaces with guaranteed quality, users can control the level of quality of the resulting triangulation that their application problems require. We have implemented and tested the proposed algorithm for random non-intersecting spheres, and the experimental result shows the proposed algorithm produces quality triangulations on Voronoi surfaces satisfying the quality criterion.