  Dissipation-free current transport in type II superconductors requires
vortices to be pinned by defects in the underlying material. The pinning
capacity of a defect is quantified by the Labusch parameter $\kappa \sim
f_p/\xi\bar{C}$, measuring the pinning force $f_p$ relative to the elasticity
$\bar{C}$ of the vortex lattice, with $\xi$ denoting the coherence length (or
vortex core size) of the superconductor. The critical value $\kappa = 1$
separates weak from strong pinning, with a strong defect at $\kappa > 1$ able
to pin a vortex on its own. So far, this weak-to-strong pinning transition has
been studied for isotropic defect potentials, resulting in a critical exponent
$\mu = 2$ for the onset of the strong pinning force density $F_\mathrm{pin}
\sim n_p f_p (\xi/a_0)^2(\kappa-1)^\mu$, with $n_ p$ denoting the density of
defects and $a_0$ the intervortex distance. The behavior changes dramatically
when studying anisotropic defects with no special symmetries: the strong
pinning then originates out of isolated points with length scales growing as
$\xi (\kappa - 1)^{1/2}$, resulting in a different force exponent $\mu = 5/2$.
Our analysis of the strong pinning onset for arbitrary defect potentials
$e_p(\mathbf{R})$, with $\mathbf{R}$ a planar coordinate, makes heavy use of
the Hessian matrix describing its curvature and leads us to interesting
geometrical structures. Both, onset and merger points are defined by local
differential properties of the Hessian's determinant $D(\mathbf{R})$,
specifically, its minima and saddle points. Extending our analysis to the case
of a random two-dimensional pinning landscape, we discuss the topological
properties of unstable and bistable regions as expressed through the Euler
characteristic, with the latter related to the local differential properties of
$D(\mathbf{R})$ through Morse theory.
