  Some new directions to lay a rigorous mathematical foundation for the
phase-portrait-based modelling of fingerprints are discussed in the present
work. Couched in the language of dynamical systems, and preparing to a
preliminary modelling, a back-to-basics analogy between Poincar\'{e}'s
categories of equilibria of planar differential systems and the basic
fingerprint singularities according to Purkyn\v{e}-Galton's standards is first
investigated. Then, the problem of the global representation of a fingerprint's
flow-like pattern as a smooth deformation of the phase portrait of a
differential system is addressed. Unlike visualisation in fluid dynamics, where
similarity between integral curves of smooth vector fields and flow streamline
patterns is eye-catching, the case of an oriented texture like a fingerprint's
stream of ridges proved to be a hard problem since, on the one hand, not all
fingerprint singularities and nearby orientational behaviour can be modelled by
canonical phase portraits on the plane, and on the other hand, even if it were
the case, this should lead to a perplexing geometrical problem of connecting
local phase portraits, a question which will be formulated within
Poincar\'{e}'s index theory and addressed via a normal form approach as a
bivariate Hermite interpolation problem. To a certain extent, the material
presented herein is self-contained and provides a baseline for future work
where, starting from a normal form as a source image, a transport via large
deformation flows is envisaged to match the fingerprint as a target image.
