Visualizing Critical Points of Arbitrary Poincaré -Index

One major discipline of Scientific Visualization is the display of vector field data. We address the problem of visualizing vector fields containing critical points of higher order Poincaré-Index. We use polynomial pieces so that the degree is high enough to get the right Poincaré-Index at the location of the critical points. This is an essential improvement because nearly all visualization of vector fields up to now used piecewise linear interpolation and therefore can not show this kind of critical points. A description of vector fields in Clifford algebra opens the way to a better understanding of some plane polynomial vector fields. This is used to choose the polynomial approximations needed for the visualization.