  The aim of this article is twofold. First, we show the evolution of the
vortex filament equation (VFE) for a regular planar polygon in the hyperbolic
space. Unlike in the Euclidean space, the planar polygon is open and both of
its ends grow exponentially, which makes the problem more challenging from a
numerical point of view. However, with fixed boundary conditions, a finite
difference scheme and a fourth-order Runge--Kutta method in time, we show that
the numerical solution is in complete agreement with the one obtained from
algebraic techniques. Second, as in the Euclidean case, we claim that, at
infinitesimal times, the evolution of VFE for a planar polygon as the initial
datum can be described as a superposition of several one-corner initial data.
As a consequence, not only can we compute the speed of the center of mass of
the planar polygon, but the relationship also allows us to compare the time
evolution of any of its corners with that in the Euclidean case.
