Algebraic Geometry 3 Curves Jaobians by Parshin Shafarevich

By Parshin Shafarevich

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The original flow field. (b) Scale-space lifetime of the critical points in the interval τ = 0 . . 10 is computed by our algorithm. 3) and the remaining points have been used to seed stream lines in their vicinity. done in only 24 seconds. 3 does significantly reduce the amount of streamlines that are displayed and accordingly the problems with visual clutter, but yet one can still clearly discern the overall behavior of the flow. Again, both tracking approaches produce comparable results. 7 Conclusion and Future Work In this paper we have presented a first approach on a tracking algorithm for vector field singularities in scale-space that uses explicit knowledge of the evolution of the field along the scale-axis.

J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer Verlag, 1986. 7. J. L. Helman and L. Hesselink. Representation and Display of Vector Field Topology in Fluid Flow Data Sets. IEEE Computer, 22(8):27–36, 1989. 8. J. L. Helman and L. Hesselink. Visualizing Vector Field Topology in Fluid Flows. IEEE Comput. Graph. , 11(3):36–46, 1991. 9. T. Iijima. Basic theory on normalization of a pattern (in case of typical onedimensional pattern).

Fig. 11 shows a slice of the resulting voxel field after tracing the particle for 109 time steps. Its resolution is 750 × 600 × 600 and it spans the complete 32 Ronald Peikert and Filip Sadlo bubble. Fig. 12 shows a finer sampling of a subregion. This makes the massive folding of the surface visible. Fig. 13 and 14 show an isosurface of the voxel field with the complete bubble. The isolevel was chosen to be 5% instead of 50% in order to avoid unmanageably many triangles in the fine folds. By adding a Gaussian smoothing step, we were able to cut down the the triangle count to about 20 million.

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