  Lagrangian Descriptors (LDs) are scalar quantities able to reveal
separatrices, manifolds of hyperbolic saddles, and chaotic seas of dynamical
systems. A popular version of the LDs consists in computing the arc-length of
trajectories over a calibrated time-window. Herein we introduce and exploit an
intrinsic geometrical parametrisation of LDs, free of the time variable, for 1
degree-of-freedom Hamiltonian systems. The parametrisation depends solely on
the energy of the system and on the geometry of the associated level curve. We
discuss applications of this framework on classical problems on the plane and
cylinder, including the cat's eye, 8-shaped and fish-tail separatrices. The
developed apparatus allows to characterise semi-analytically the rate at which
the derivatives of the geometrical LDs become singular when approaching the
separatrix. For the problems considered, the same power laws of divergence are
found irrespective from the dynamical system. Some of our results are connected
with existing estimates obtained with the temporal LDs under approximations.
The geometrical formalism provides alternative insights of the mechanisms
driving this dynamical indicator.
