  We present a novel approach to finding critical points in cell-wise
barycentrically or bilinearly interpolated vector fields on surfaces. The
Poincar\e index of the critical points is determined by investigating the
qualitative behavior of 0-level sets of the interpolants of the vector field
components in parameter space using precomputed combinatorial results, thus
avoiding the computation of the Jacobian of the vector field at the critical
points in order to determine its index. The locations of the critical points
within a cell are determined analytically to achieve accurate results. This
approach leads to a correct treatment of cases with two first-order critical
points or one second-order critical point of bilinearly interpolated vector
fields within one cell, which would be missed by examining the linearized field
only. We show that for the considered interpolation schemes determining the
index of a critical point can be seen as a coloring problem of cell edges. A
complete classification of all possible colorings in terms of the types and
number of critical points yielded by each coloring is given using computational
group theory. We present an efficient algorithm that makes use of these
precomputed classifications in order to find and classify critical points in a
cell-by-cell fashion. Issues of numerical stability, construction of the
topological skeleton, topological simplification, and the statistics of the
different types of critical points are also discussed.
