The study of planar and spherical geometric subdivision schemes was done in Dyn and Hormann (2012); Bellaihou and Ikemakhen (2020). In this paper we complete this study by examining the hyperbolic case. We define general interpolatory geometric subdivision schemes generating curves on the hyperbolic plane by using geodesic polygons and the hyperbolic trigonometry. We show that a hyperbolic interpolatory geometric subdivision scheme is convergent if the sequence of maximum edge lengths is summable and the limit curve is $G^1$-continuous if in addition the sequence of maximum angular defects is summable. In particular, we study the case of bisector interpolatory schemes. Some examples are given to demonstrate the properties of these schemes and some fascinating images on Poincaré disk are produced from these schemes.

Dyn 和 Hormann (2012) 对平面和球形几何细分方案进行了研究；Bellaihou 和 Ikemakhen (2020)。在本文中，我们通过检查双曲线情况来完成这项研究。我们通过使用测地线多边形和双曲三角学定义了在双曲平面上生成曲线的一般插值几何细分方案。我们表明，如果最大边长的序列是可求和的，并且极限曲线为，则双曲插值几何细分方案是收敛的$G^1$-如果此外最大角度缺陷的序列是可求和的，则连续。我们特别研究了平分线插值方案的情况。给出了一些例子来证明这些方案的特性，并且从这些方案中产生了一些庞加莱圆盘上的迷人图像。