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.

翻译：文章的目的是双重的。 首先,我们展示了超双曲空间中常规平面多边形的涡旋丝形方程式(VFE)的演变过程。 第二,与欧clidean案不同的是,平面多边形是开放的,其两端都成倍增长,这使得问题从数字角度更加具有挑战性。然而,在固定边界条件、有限差异办法和第四阶龙格-库塔方法的情况下,我们显示数字解决办法与从代数技术中获得的解决办法完全一致。第二,我们声称,在极小的时期,作为初始数据,对平面多边形的VFE的演变可以被描述为数个一角初步数据的超级位置。因此,我们不仅能够计算计划多边形中心的质量速度,而且这种关系也使我们能够将其任何角落的进化速度与欧clidean案的进化速度进行比较。