Journal of Digital Contents Society (디지털콘텐츠학회 논문지)

Digital Contents Society (한국디지털콘텐츠학회)
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The Phase Space Analysis of 3D Vector Fields

3차원 벡터 필드의 위상 공간 분석

  • Jung, Il-Hong (Department of Computer Engineering, Daejeon University) ;
  • Kim, Yong Soo (Department of Computer Engineering, Daejeon University)
  • Received : 2015.12.09
  • Accepted : 2015.12.28
  • Published : 2015.12.31
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Abstract

This paper presents a method to display the 3D vector fields by analyzing phase space. This method is based on the connections between ordinary differential equations and the topology of vector fields. The phase space analysis should be geometric interpolation of an autonomous system of equation in the form of the phase space. Every solution of it system of equations corresponds not to a curve in a space, but the motion of a point along the curve. This analysis is the basis of this paper. This new method is required to decompose the hexahedral cell into five or six tetrahedral cells for 3D vector fields. The critical points can be easily found by solving a simple linear system for each tetrahedron. The tangent curves can be integrated by finding the intersection points of an integral curve traced out by the general solution of each tetrahedron and plane containing a face of the tetrahedron.

본 논문에서는 위상 공간 분석을 통해 3D 벡터 필드를 표현하는 방법을 제안한다. 이 방법은 상미분 방정식과 벡터 필드 위상과의 연결에 기초를 두고 있다. 위상 공간 분석은 위상 공간 형태의 자율 방정식 시스템의 기하학적 보간법이 되어야 한다. 이 방정식 시스템의 모든 해는 공간에서의 곡선이 아니라 곡선을 따라가는 점의 움직임과 일치한다. 이러한 분석은 이 논문의 기반이다. 새로운 방법은 3차원 벡터필드에서 육면체 셀을 5 또는 6개의 사면체 셀로 분해하는 것을 요구한다. 임계점은 각 사면체의 간단한 선형 시스템을 풀어서 간단하게 구할 수 있다. 각 사면체의 일반해에 의해 그려지는 전체 곡선과 사면체의 한 면을 포함하는 평면과의 교차점을 계산함으로써 탄젠트 곡선은 구해진다.

Keywords

References

  1. I. -H. Jung, "A Tetrahedral Decomposition Method for Computing Tangent Curves of 3D Vector Fields", Journal of Digital Contents Society, Vol.16, No.41, pp.575-581, Aug. 2015. https://doi.org/10.9728/dcs.2015.16.4.575
  2. I. -H. Jung, "An Efficient Visualization Method of Two-Dimensional Vector Fields", Journal of Korea Multimedia Society, Vol.12, No.11, pp.1623-1628, Nov. 2009.
  3. A. Globus, C. Levit, and T. Lasinski, "A tool for Visualizing the Topology of Three-dimensional Vector Fields," Proc. Visualization '91, IEEE Computer Society, pp.33-40, 1991.
  4. J. L. Helman and L. Hesselink, "Representation and Display of Vector Field Topology in Fluid Data Sets," IEEE Computer, Vol.22, No.8, pp.27-36, Aug. 1989.
  5. J. L. Helman and L. Hesselink, "Surface Presentation of Two- and Three-Dimensional Fluid Flow Topology," Proc. Visualization '90, IEEE Computer Society, pp.6-13, 1990.
  6. J. L. Helman and L. Hesselink, "Visualizing Vector Field Topology in Fluid Flows," IEEE Computer Graphics and Applications, Vol.11, pp.36-46, May 1991. https://doi.org/10.1109/38.79452
  7. J. C. Kim, K. O. Kim, E. K. Kim and C. Y. Kim, "3D Visualization System for Realtime Environmental Data", Journal of Digital Contents Society, Vol.9, No.4, pp.707-715, Dec. 2008.
  8. V. I. Arnold, "Ordinary Differential Equations", MIT Press, 1973.
  9. S. Birkhoff and G. Rota, "Ordinary Differential Equations", fourth edition, Wiley, NewYork, 1989.
  10. G. M. Nielson, I. -J. Jung, N. Srinivasan, J, Sung and J. -B. Yoon, "Tools for Computing Tangent Curves and Topological Graphs for Visualizing Piecewise Linearly Varying Vector Fields over Triangulate d Domains," in: Scientific Visualization: Overview, Methodologies and Techniques, G. Nielson, H. Hagen and H. Muller, editors, IEEE Computer Society Press, Los Alamitos, CA, pp.527-562, 1997.
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterlin, and B. P. Flannery, "Numerical Recipes in C: The Art of Scientific Computing", Second Edition, Cambridge University Press, Cambridge, 1992
  12. G. M. Nielson, "Visualization in Scientific and Engineering Computation," IEEE Computer, Vol.24, No.9, pp.58-66, 1991.
  13. G. M. Nielson and A. Kaufman, "Visualization Graduates," IEEE Computer Graphics and Applications, Vol.14, No. 5, pp.17-18, Sep. 1994.
  14. S. -K. Ueng, C. Sikorski, and K.-L Ma, "Efficient Streamline, Streamribbon, and Streamtube Constructions on Unstructured Grids," IEEE Transactions of Visualization and Computer Graphics, Vol.2, No.2, pp.100-108, Jun. 1996. https://doi.org/10.1109/2945.506222

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