• 제목/요약/키워드: Voronoi

검색결과 200건 처리시간 0.019초

Divide and conquer algorithm for a voronoi diagram of simple curves

  • Kim, Deok-Soo;Hwang, Il-Kyu;Park, Bum-Joo
    • 한국경영과학회:학술대회논문집
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    • 대한산업공학회/한국경영과학회 1994년도 춘계공동학술대회논문집; 창원대학교; 08월 09일 Apr. 1994
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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.

Voronoi diagrams, quasi-triangulations, and beta-complexes for disks in R2: the theory and implementation in BetaConcept

  • Kim, Jae-Kwan;Cho, Youngsong;Kim, Donguk;Kim, Deok-Soo
    • 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).

점 집합의 보로노이 다이어그램을 이용한 원 집합의 보로노이 다이어그램의 계산: I. 위상학적 측면 (The Computation of the Voronoi Diagram of a Circle Set Using the Voronoi Diagram of a Point Set: I. Topology)

  • 김동욱;김덕수;조동수
    • 한국CDE학회논문집
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    • 제6권1호
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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.

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미지의 환경에서 트리구조를 이용한 군집로봇의 분산 탐색 (Distributed Search of Swarm Robots Using Tree Structure in Unknown Environment)

  • 이기수;주영훈
    • 전기학회논문지
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    • 제67권2호
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    • pp.285-292
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    • 2018
  • In this paper, we propose a distributed search of a cluster robot using tree structure in an unknown environment. In the proposed method, the cluster robot divides the unknown environment into 4 regions by using the LRF (Laser Range Finder) sensor information and divides the maximum detection distance into 4 regions, and detects feature points of the obstacle. Also, we define the detected feature points as Voronoi Generators of the Voronoi Diagram and apply the Voronoi diagram. The Voronoi Space, the Voronoi Partition, and the Voronoi Vertex, components of Voronoi, are created. The generated Voronoi partition is the path of the robot. Voronoi vertices are defined as each node and consist of the proposed tree structure. The root of the tree is the starting point, and the node with the least significant bit and no children is the target point. Finally, we demonstrate the superiority of the proposed method through several simulations.

삼차원 구의 보로노이 다이어그램 계산을 위한 두 가지 알고리듬 및 단백질구조채석에의 응용 (Two Algorithms for Constructing the Voronoi Diagram for 3D Spheres and Applications to Protein Structure Analysis)

  • 김동욱;조영송;김덕수
    • 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.

딜러니 개선 알고리듬을 이용한 삼차원 구의 보로노이 곡면 삼각화 (Triangulation of Voronoi Faces of Sphere Voronoi Diagram using Delaunay Refinement Algorithm)

  • 김동욱
    • 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.

영역 확장법을 통한 평면에서 원들의 보로노이 다이어그램의 강건한 계산 (Robust Construction of Voronoi Diagram of Circles by Region-Expansion Algorithm)

  • 김동욱
    • 산업경영시스템학회지
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    • 제42권3호
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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.

Topology Representation for the Voronoi Diagram of 3D Spheres

  • Cho, Young-Song;Kim, Dong-Uk;Kim, Deok-Soo
    • 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.

원 집합의 보로노이 다이어그램을 구하는 모서리 플립 알고리듬 (Edge-flipping algorithm for computing the circle set Voronoi diagram)

  • 김동욱;김덕수
    • 한국산업경영시스템학회:학술대회논문집
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    • 한국산업경영시스템학회 2002년도 춘계학술대회
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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.

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