The Community for Technology Leaders
RSS Icon
Issue No.04 - July/August (2009 vol.15)
pp: 682-695
Allen Van Gelder , University of California, Santa Cruz
Alex Pang , University of California, Santa Cruz
A new method for finding the locus of parallel vectors is presented, called PVsolve. A parallel-vector operator has been proposed as a visualization primitive, as several features can be expressed as the locus of points where two vector fields are parallel. Several applications of the idea have been reported, so accurate and efficient location of such points is an important problem. Previously published methods derive a tangent direction under the assumption that the two vector fields are parallel at the current point in space, then extend in that direction to a new point. PVsolve includes additional terms to allow for the fact that the two vector fields may not be parallel at the current point, and uses a root-finding approach. Mathematical analysis sheds new light on the feature flow field technique (FFF) as well. The root-finding property allows PVsolve to use larger step sizes for tracing parallel-vector curves, compared to previous methods, and does not rely on sophisticated differential equation techniques for accuracy. Experiments are reported on fluid flow simulations, comparing FFF and PVsolve.
Parallel vectors, feature flow field, vortex core, flow visualization, PVsolve, adjugate matrix, Newton-Raphson root finding, dimensionless projection vector.
Allen Van Gelder, Alex Pang, "Using PVsolve to Analyze and Locate Positions of Parallel Vectors", IEEE Transactions on Visualization & Computer Graphics, vol.15, no. 4, pp. 682-695, July/August 2009, doi:10.1109/TVCG.2009.11
[1] D.C. Banks and B.A. Singer, “Vortex Tubes in Turbulent Flows: Identification, Representation, Reconstruction,” Proc. IEEE Visualization Conf. '94, pp.132-139, 1994.
[2] D.C. Banks and B.A. Singer, “A Predictor-Corrector Technique for Visualizing Unsteady Flow,” IEEE Trans. Visualization and Computer Graphics, vol. 1, no. 2, pp.151-163, June 1995.
[3] C. Garth, R.S. Laramee, X. Tricoche, J. Schneider, and H. Hagen, “Extraction and Visualization of Swirl and Tumble Motion from Engine Simulation Data,” Topology-Based Methods in Visualization, H. Hauser, H. Hagen, and H. Theisel, eds., pp.121-135, Springer, 2007.
[4] G.H. Golub and C.F. Van Loan, Matrix Computations, third ed. Johns Hopkins University Press, 1996.
[5] A.J. Hanson, Geometry for N-Dimensional Graphics, P. Heckbert, ed., pp.149-170, Academic Press, 1994.
[6] M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, 1974.
[7] R. Hooke and T.A. Jeeves, “‘Direct Search’ Solution of Numerical and Statistical Problems,” J. ACM, vol. 8, no. 2, pp.212-229, 1961.
[8] J. Jeong and F. Hussain, “On the Identification of a Vortex,” J.Fluid Mechanics, vol. 285, pp.69-94, 1995.
[9] Kitware, ParaView Guide, Kitware, Version 3, http:/www., Feb. 2008.
[10] M. Marcus and H. Minc, Introduction to Linear Algebra. MacMillan, 1969.
[11] W.S. Massey, “Cross Products of Vectors in Higher Dimensional Euclidean Spaces,” Am. Math. Monthly, vol. 90, pp.697-701, 1983.
[12] G.M. Nielson and I.-H. Jung, “Tools for Computing Tangent Curves for Linearly Varying Vector Fields over Tetrahedral Domains,” IEEE Trans. Visualization and Computer Graphics, vol. 5, no. 4, pp.360-372, Oct.-Dec. 1999.
[13] R. Peikert and M. Roth, “The Parallel Vector Operator—A Vector Field Visualization Primitive,” Proc. IEEE Visualization Conf. '99, pp.263-270, 1999.
[14] M. Roth, “Automatic Extraction of Vortex Core Lines and Other Line-Type Features for Scientific Visualization,” PhD thesis, ETH Zurich, Inst. of Scientific Computing, Diss. ETH No. 13673, 2000.
[15] L.F. Shampine and M.W. Reichelt, “The Matlab ODE Suite,” SIAM J. Scientific Computing, vol. 18, pp.1-22, 1997.
[16] S. Stegmaier, U. Rist, and T. Ertl, “Opening The Can of Worms: An Exploration Tool for Vortical Flows,” Proc. IEEE Visualization Conf. '05, pp.463-470, 2005.
[17] G.W. Stewart, “On the Adjugate Matrix,” Linear Algebra and Its Applications, vol. 283, pp.151-164, 1998.
[18] J. Sukharev, X. Zheng, and A. Pang, “Tracing Parallel Vectors,” Proc. SPIE Visual Data Analysis and Exploration, 2006.
[19] H. Theisel, J. Sahner, T. Weinkauf, H.-C. Hege, and H.-P. Seidel, “Extraction of Parallel Vector Surfaces in 3D Time-Dependent Fields and Application to Vortex Core Line Tracking,” Proc. Visualization Conf. '05, pp.631-638, 2005.
[20] H. Theisel and H.-P. Seidel, “Feature Flow Fields,” Proc. Joint Eurographics—IEEE TCVG Symp. Visualization (VisSym'03), pp. 141-148, 2003.
[21] A. Van Gelder, “Report on Relaxed Jordan Canonical Form for Computer Animation and Visualization,” technical report, Univ. of California, Santa Cruz, , Jan. 2009.
[22] T. Weinkauf, “Extraction of Topological Structures in 2D and 3D Vector Fields,” PhD thesis, Otto von Guericke Univ. of Magdeburg, Germany, 2008.
[23] T. Weinkauf, J. Sahner, H. Theisel, H.-C. Hege, and H.-P. Seidel, “A Unified Feature Extraction Architecture,” Active Flow Control, R. King, ed., pp.119-133, Springer, 2007.
19 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool