Hyperbolic interpolatory geometric subdivision schemes

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 G1-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. •We define interpolatory geometric subdivision schemes on the hyperbolic plane.•We derive sufficient conditions for convergence and G1-Continuity.•We give two examples of hyperbolic geometric subdivision schemes.•Some fascinating images on Poincaré disk are produced from these schemes.