• International Journal of CAD/CAM
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• 제5권1호
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• pp.59-68
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• 2005

Euclidean Voronoi diagram of spheres in 3-dimensional space has not been explored as much as it deserves even though it has significant potential impacts on diverse applications in both science and engineering. In addition, studies on the data structure for its topology have not been reported yet. Presented in this, paper is the topological representation for Euclidean Voronoi diagram of spheres which is a typical non-manifold model. The proposed representation is a variation of radial edge data structure capable of dealing with the topological characteristics of Euclidean Voronoi diagram of spheres distinguished from those of a general non-manifold model and Euclidean Voronoi diagram of points. Various topological queries for the spatial reasoning on the representation are also presented as a sequence of adjacency relationships among topological entities. The time and storage complexities of the proposed representation are analyzed.

• 한국CDE학회논문집
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• 제11권2호
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• pp.97-106
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• 2006

Voronoi diagrams have been known for numerous important applications in science and engineering including CAD/CAM. Especially, the Voronoi diagram for 3D spheres has been known as very useful tool to analyze spatial structural properties of molecules or materials modeled by a set of spherical atoms. In this paper, we present two algorithms, the edge-tracing algorithm and the region-expansion algorithm, for constructing the Voronoi diagram of 3D spheres and applications to protein structure analysis. The basic scheme of the edge-tracing algorithm is to follow Voronoi edges until the construction is completed in O(mn) time in the worst-case, where m and n are the numbers of edges and spheres, respectively. On the other hand, the region-expansion algorithm constructs the desired Voronoi diagram by expanding Voronoi regions for one sphere after another via a series of topology operations, starting from the ordinary Voronoi diagram for the centers of spheres. It turns out that the region-expansion algorithm also has the worst-case time complexity of O(mn). The Voronoi diagram for 3D spheres can play key roles in various analyses of protein structures such as the pocket recognition, molecular surface construction, and protein-protein interaction interface construction.

• 한국경영과학회:학술대회논문집
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• 한국경영과학회/대한산업공학회 2005년도 춘계공동학술대회 발표논문
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• pp.842-847
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• 2005

Although there are many applications of Euclidean Voronoi diagram for spheres in a 3D space in various disciplines from sciences and engineering, it has not been studied as much as it deserves. In this paper, we present an edge-tracing algorithm to compute the Euclidean Voronoi diagram of 3-dimensional spheres in O(mn) in the worst-case, where m is the number of edges of Voronoi diagram and n is the number of spheres. After building blocks for the algorithm, we show an example of Voronoi diagram for atoms using actual protein data and discuss its applications for protein analysis.

• 산업경영시스템학회지
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• 제35권4호
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• pp.235-243
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• 2012

The Voronoi diagram of spheres and power diagram have been known as powerful tools to analyze spatial characteristics of weighted points, and these structures have variety range of applications including molecular spatial structure analysis, location based optimization, architectural design, etc. Due to the fact that both diagrams are based on different distance metrics, one has better usability than another depending on application problems. In this paper, we compare these diagrams in various situations from the user's viewpoint, and show the Voronoi diagram of spheres is more effective in the problems based on the Euclidean distance metric such as nearest neighbor search, path bottleneck locating, and internal void finding.

• 산업경영시스템학회지
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• 제41권4호
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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 Computational Design and Engineering
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• 제1권2호
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• pp.79-87
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• 2014

Voronoi diagrams are powerful for solving spatial problems among particles and have been used in many disciplines of science and engineering. In particular, the Voronoi diagram of three-dimensional spheres, also called the additively-weighted Voronoi diagram, has proven its powerful capabilities for solving the spatial reasoning problems for the arrangement of atoms in both molecular biology and material sciences. In order to solve application problems, the dual structure, called the quasi-triangulation, and its derivative structure, called the beta-complex, are frequently used with the Voronoi diagram itself. However, the Voronoi diagram, the quasi-triangulation, and the beta-complexes are sometimes regarded as somewhat difficult for ordinary users to understand. This paper presents the two-dimensional counterparts of their definitions and introduce the BetaConcept program which implements the theory so that users can easily learn the powerful concept and capabilities of these constructs in a plane. The BetaConcept program was implemented in the standard C++ language with MFC and OpenGL and freely available at Voronoi Diagram Research Center (http://voronoi.hanyang.ac.kr).

• International Journal of CAD/CAM
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• 제7권1호
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• pp.91-101
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• 2007

To understand the structure of molecules, various computational methodologies have been extensively investigated such as the Voronoi diagram of the centers of atoms in molecule and the power diagram for the weighted points where the weights are related to the radii of the atoms. For a more improved efficiency, constructs like an $\alpha$-shape or a weighted $\alpha$-shape have been developed and used frequently in a systematic analysis of the morphology of molecules. However, it has been recently shown that $\alpha$-shapes and weighted $\alpha$-shapes lack the fidelity to Euclidean distance for molecules with polysized spherical atoms. We present the theory as well as algorithms of $\beta$-shape and $\beta$-complex in $\mathbb{R}^3$ which reflects the size difference among atoms in their full Euclidean metric. We show that these new concepts are more natural for most applications and therefore will have a significant impact on applications based on particles, in particular in molecular biology. The theory will be equivalently useful for other application areas such as computer graphics, geometric modeling, chemistry, physics, and material science.

• 정보처리학회논문지A
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• 제19A권4호
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• pp.169-174
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• 2012

본 논문에서는 주어진 단백질 분자에 대해 3차원 보로노이 다이어그램을 계산하는 알고리즘을 제안한다. 분자는 반경이 서로 다른 구의 집합으로 표현되며, 각 구의 반경은 원자의 반데르바스 (van der Waals) 반경에 대응한다. 보로노이 다이어그램은 3차원 공간을 복셀(voxel)의 집합으로 분할한 뒤, 보로노이 다이어그램을 포함하는 복셀을 보수적으로 추출함으로써 구성된다. 분자의 계층적 성질을 이용하여 BVH(bounding volume hierarchy)를 구성하고, CUDA 프로그래밍을 통하여 그래픽 하드웨어 가속을 활용함으로써 계산 시간 효율성을 높인다. 공간이 최대 $2^{24}$개의 복셀로 분할될 경우, 단일 코어 CPU로 구현하는 알고리즘에 비해 계산 속도가 323배 가량 향상 되었다.