• Journal of the Chungcheong Mathematical Society
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• v.34 no.4
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• pp.387-395
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• 2021

In this paper, we give a characterization of a Delaunay surface in ℝ³. Let Σ be a CMC-H surface in ℝ³ with H ≠ 0. If Σ meets a plane with constant contact angle along a circle, then it is rotationally symmetric, i.e., Σ is part of a Delaunay surface.

• 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.

• Journal of the Korean Society for Precision Engineering
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• v.17 no.7
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• pp.181-188
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• 2000

The objective of this research is to develop an efficient reverse engineering method of 3-dimensional compound surfaces for raped prototyping process. As a first step, several image processing techniques were applied to the measured data obtained using laser scanner. And the boundary information of the compound surface were extracted to divide the surface into several separate regions. As a next step, the Delaunay triangulation method were applied to reconstruct the surface based on the measured data and the boundary information. Finally, the STL file were created for the rapid prototyping process. Required simulations and experiments were performed and the results were analyzed to show the effectiveness of the proposed methods.

• Journal of Biomedical Engineering Research
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• v.21 no.3 s.61
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• pp.225-231
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• 2000

The Delaunay triangulation. which is one of the surface rendering methods. have been evaluated as a good algorithm which can solve the geometrical connection problems and make high quality images. However this method also have the problem that is the 'non-solid' connection between slices. In this paper, we proposed a new method for the Delaunay triangulation for the surface rendering from 2D medical images in the PC environment. The proposed method was designed to eliminate 'non-solid' tetrahedra. which had no correspondence to the geometric closeness, and used elimination algorithm with modified External Voronoi Skeleton path. This method enabled us to eliminate 'non-solid' tetrahedra without affecting other regions. Thus we could effectively reconstruct the complex shaped objects which were compactly connected with tetrahedra.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.27 no.5
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• pp.681-687
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• 2003

Terrain surfaces have to be modeled in very detail and wheel-surface contacting geometry must be well defined in order to obtain proper ground-reaction and friction forces fur realistic simulation of off-road vehicles. Delaunay triangulation is one of the most widely used methods in modeling 3-dimensional terrain surfaces, and the T-search is a relevant algorithm for searching resulting triangular polygons. The T-search method searches polygons in a successive order and may not allow real-time computation of off-road vehicle dynamics if the terrain is modeled with many polygons, depending on the computer performance used in the simulation. The dynamic T-search, which is proposed in this paper, combines conventional T-search and the concept of the dynmaic-window search which uses reduced searching windows or sets of triangular surface polygons at each frame by taking advantage of the information regarding dynamic charactereistics of a simulated vehicle. Numerical tests show improvement of searching speeds by about 5% for randomly distributed triangles. For continuous search following a vehicle path, which occurs in actual vehicle simulation, the searching speed becomes 4 times faster.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.25 no.8
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• pp.1246-1252
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• 2001

Reverse engineering has been widely used for the shape reconstruction of an object without CAD data and the measurement of clay or wood models for the development of new products. To generate a surface from measured points by a laser scanner, typical steps include the scanning of a clay or wood model and the generation of manufacturing data like STL file. A laser scanner has a great potential to get geometrical data of a model for its fast measuring speed and higher precision. The data from a laser scanner are composed of many line stripes of points. A new approach to remove point data with Delaunay triangulation is introduced to deal with problems during reverse engineering process. This approach can be used to reduce a number of measuring data from laser scanner within tolerance, thus it can avoid the time for handling point data during modelling process and the time for verifying and slicing STL model during RP process.

• Journal of the Korea Computer Graphics Society
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• v.13 no.4
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• pp.21-27
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• 2007

The 3D Delaunay triangulation is the most widely used method for the mesh creation via the triangulation of a 3D point cloud. However, the method involves a heavy computational cost and, hence, in many interactive applications, it is not appropriate for surface triangulation. In this paper, we propose an efficient triangulation method to create a surface mesh from a 3D point cloud. We divide a set of object points into multiple subsets and apply the 2D Delaunay triangulation to each subset. A given 3D point cloud is cut into slices with respect to the OBB(Oriented Bounding Box) of the point set. The 2D Delaunay triangulation is applied to each subset producing a partial triangulation. The sum of the partial triangulations constitutes the global mesh. As a postprocessing process, we eliminate false edges introduced in the split steps of the triangulation and improve the results. The proposed method can be effectively applied to various image-based modeling applications.

• Journal of Korea Game Society
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• v.2 no.1
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• pp.62-68
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• 2002

This paper presents a method for tessellating trimmed NURBS surface with preserving sharp shape Although several existing approaches need a large number of triangular meshes to represent sharp shape of surface, resulting triangular meshes may not reflect sharp edges properly. In this study, we flit detect the sharp shape of NURBS surface automatically using C1 continuous condition and then use constraint Delaunay triangulation method to present exact sharp shape with the minimum triangular meshes. And we also use approximated developed surface domain as triangulation domain of rimmed NURBS surface. In this way, the shape of triangular elements on the triangular domains is approximately preserved and can avoid distortion when mapped into three-dimensional space. finally, we show examples that demonstrate the effectiveness of the proposed scheme in terms of reducing the number of triangular meshes and preserving sharp shape of surface more exactly.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.26 no.5
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• pp.803-812
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• 2002

Reverse engineering has been widely used for the shape reconstruction of an object without CAD data and the measurement of clay or wood models for the development of new products. To generate a surface from measured points by a laser scanner, typical steps include the scanning of a clay or wood model and the generation of manufacturing data like STL file. A laser scanner has a great potential to get geometrical data of a model for its fast measuring speed and higher precision. The data from a laser scanner are composed of many line stripes of points. A new approach to remove point data with Delaunay triangulation is introduced to deal with problems during reverse engineering process. The selection of group of triangles to be triangulated based on the angle between triangles is used for robust and reliable implementation of Delaunay triangulation as preliminary steps. Developed software enables the user to specify the criteria for the selection of group of triangles either by the angle between triangles or the percentage of triangles reduced. The time and error for handling point data during modelling process can be reduced and thus RP models with accuracy will be helpful to automated process.

• The Transactions of the Korean Institute of Electrical Engineers
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• v.37 no.12
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• pp.847-853
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• 1988

A method of three-demensional finite element mesh generation is presented in this paper. This method is based on the Delaunay's triangulation whose dual is Voronoi's diagram. A set of points is given on the boundary surface of the concerning domain and the initial tetrahedra are generated by the given set of points. Then, the quality of every tetrahedron is investigated and the interior points are generated near the tetrahedra which are inferior in quality and the tetrahedra of good quality can be controlled by the density of the initial boundary points. Regions with different material constant can be refined in tetrahedra respectively. To confirm the effectiveness of this algorithm,the total volume of tetrahedra was compared to the true volume and this mesh generator was applied to a three-dimensional electostatic problem.