• Proceedings of the Korean Operations and Management Science Society Conference
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• 1994.04a
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• pp.691-700
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• 1994

Voronoi diagram of a set of geometric entities on a plane such as points, line segments, or arcs is a collection of Voronoi polygons associated with each entity, where Voronoi polygon of an entity is a locus of point which is closer to the associated entity than any other entity. Voronoi diagram is one of the most fundamental geometrical construct and well-known for its theoretical elegance and the wealth of applications. Various geometric problems can be solved with the aid of Voronoi diagram. For example, the maximum tool diameter of a milling cutter for rough cutting in a pocket can be easily found, and the pocketing tool path can be efficiently generated from Voronoi diagram. In PCB design, the design rule checking can be easily done via Voronoi diagram, too. This paper discusses an algorithm to construct Voronoi diagram of a simple polygon which consists of simple curves such as line segments as well as arcs in a plane with O(nlogn) time complexity by employing the divide and conquer scheme.

• Journal of Computational Design and Engineering
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• v.1 no.2
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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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• v.5 no.1
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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.

• Korean Journal of Computational Design and Engineering
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• v.6 no.1
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• pp.24-30
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• 2001

An efficient and robust algorithm to compute the exact Voronoi diagram of a circle set is presented. The circles are located in a two dimensional Euclidean space, the radii of the circles are non-negative and not necessarily equal, and the circles are allowed to intersect each other. The idea of the algorithm is to use the topology of the point set Voronoi diagram as a seed so that the correct topology of the circle set Voronoi diagram can be obtained through a number of edge flipping operations. Then, the geometries of the Voronoi edges of the circle set Voronoi diagram are computed. In particular, this paper discusses the topological aspect of the algorithm, and the following paper discusses the geometrical aspect. The main advantages of the proposed algorithm are in its robustness, speed, and the simplicity in its concept as well as implementation. Since the algorithm is based on the result of the point set Voronoi diagram and the flipping operation is the only topological operation, the algorithm is always as stable as the Voronoi diagram construction algorithm of a point set.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.42 no.3
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• pp.52-60
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• 2019

This paper presents a numerically robust algorithm to construct a Voronoi diagram of circles in the plane. The circles are allowed to have intersections among them, but one circle cannot fully contain another circle. The Voronoi diagram is a tessellation of the plane into Voronoi regions of given circles. Each circle has its Voronoi region which is defined by a set of points in the plane closer to the circle than any other circles. The distance from a point p to a circle $c_i$ of center $p_i$ and radius $r_i$ is ${\parallel}p-p_i{\parallel}-r_i$, which is the closest Euclidean distance from p to the circle boundary. The proposed algorithm first constructs the point Voronoi diagram of centers of given circles, then it enlarges each point to the circle and expands its Voronoi region accordingly. This region-expansion process is done by local modifications and after completing this process for the whole circles the desired circle Voronoi diagram can be obtained. The proposed algorithm is numerically robust and we provide with a few examples to show its robustness. The algorithm runs in $O(n^2)$ time in the worst case and O(n) time on average where n is the number of the circles. The experiment shows that the region-expansion algorithm is robust and runs fast with strong linear time behavior.

• Proceedings of the Society of Korea Industrial and System Engineering Conference
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• 2002.05a
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• pp.467-472
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• 2002

Presented in this paper is an algorithm to compute the Voronoi diagram of a circle set from the Voronoi diagram of a point set. The circles are located in Euclidean plane, the radii of the circles are non-negative and not necessarily equal, and the circles are allowed to intersect each other. The idea of the algorithm is to use the topology of the point set Voronoi diagram as a seed so that the correct topology of the circle set Voronoi diagram can be obtained through a number of edge flipping operations. Then, the geometries of the Voronoi edges of the circle set Voronoi diagram are computed. The main advantages of the proposed algorithm are in its robustness, speed, and the simplicity in its concept as well as implementation.

• Korean Journal of Computational Design and Engineering
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• v.14 no.5
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• pp.306-313
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• 2009

This paper presents the unique formation process of a volumetric space with the digital algorithm developed for Voronoi diagram in order to generate an effective parametric architectural form. By applying systematic parameters of architectural conditions within digital parametric tools, the interactions among sub-spaces developed by Voronoi diagram are enhanced by manipulating the spatial structures. In this paper, we discuss how the parametric distributing and zoning geometrical system can support designers in developing a free-formed space, and research on how this system creates a 3D volumetric space. With the in-depth research on the system and structure of Voronoi diagram, the approaches to the application of Voronoi diagram into architectural form generation are clarified to be an effective, creative and successful digital tool. The result of the application of the Voronoi diagram improves the design quality with systematic language in the sense that the sub-regions are created and controlled under the systematic and balanced hierarchy having dynamic relationships among each others with the restoration of the equilibrium of forces and tensions. This 3-dimensional Voronoi diagram provides another means for designers to solve architectural issues and to reinforce their design concepts.

• Korean Journal of Computational Design and Engineering
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• v.11 no.2
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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.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 2005.05a
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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.

• Journal of the Korea Society of Computer and Information
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• v.13 no.1
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• pp.11-20
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• 2008