Abstract Integral lines of the gradient flow are standard features in continuously differentiable scalar fields that enjoy some useful properties: They cover the domain densely, do not split, merge, or intersect, and are therefore invertible. For widely used discretizations of scalar fields, the corresponding polygonal approximations of integral lines do not enjoy these properties anymore. We analyze conditions for integral lines in 2D piecewise linear (PL) scalar fields to be invertible by identifying and classifying critical edges in the underlying triangulation. We show that under mild conditions, every 2D PL scalar field can be transformed into an arbitrarily close PL field with invertible integral lines. We present an algorithm that computes this transformation and apply it to a number of test data sets.
Erxleben et al. (2026) studied this question.