For my studies, I’m aiming to implement a constrained Delaunay triagulation in the Poincare disk, using CGAL. In my case, the definition for hyperbolic Delaunay triagulation is just that all simplices are hyperbolic, not only those whose circumscribing circle is contained in the Poincare disk (see this definition [1]). So there shouldn’t be isolated edges (see this post [2]).

A comment under the linked post [2] says that: “In this case we can use the Euclidean Delaunay triangulation underlying the hyperbolic one. Then for pictures, you only need to replace Euclidean line segments by hyperbolic segments.”
I would be grateful if someone could elaborate on this more concretly.