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We present a novel approach to finding critical points in cell-wisebarycentrically or bilinearly interpolated vector fields on surfaces. ThePoincar\e index of the critical points is determined by investigating thequalitative behavior of 0-level sets of the interpolants of the vector fieldcomponents in parameter space using precomputed combinatorial results, thusavoiding the computation of the Jacobian of the vector field at the criticalpoints in order to determine its index. The locations of the critical pointswithin a cell are determined analytically to achieve accurate results. Thisapproach leads to a correct treatment of cases with two first-order criticalpoints or one second-order critical point of bilinearly interpolated vectorfields within one cell, which would be missed by examining the linearized fieldonly. We show that for the considered interpolation schemes determining theindex of a critical point can be seen as a coloring problem of cell edges. Acomplete classification of all possible colorings in terms of the types andnumber of critical points yielded by each coloring is given using computationalgroup theory. We present an efficient algorithm that makes use of theseprecomputed classifications in order to find and classify critical points in acell-by-cell fashion. Issues of numerical stability, construction of thetopological skeleton, topological simplification, and the statistics of thedifferent types of critical points are also discussed.

Finding and Classifying Critical Points of 2D Vector Fields: A Cell-Oriented Approach Using Group Theory