A classical theorem of Heinze states that negatively curved homogeneous spaces are exactly the solvable Lie groups that split as nilpotent - by - real. The cosets of the nilpotent subgroup, horospheres, carry intrinsic metrics that are exponentially distorted with respect to the metric of the whole space. Moreover, the boundary of the negatively curved space can be identified with nilpotent group together with a point at infinity. As a matter of folklore, hyperbolic groups (discrete spaces that carry homogeneous metrics with some notion of negative curvature) are thought to enjoy similar properties: exponential distortion of horospheres, an analogy between the horospheres and the boundary, and so on. In this talk, I will discuss my work to formalize these properties in the setting of hyperbolic right-angled Coxeter groups. I will also describe a set of efficient algorithms I have devised that draw pictures of large finite portions of these horospheres. The computational aspect of this work is joint with N. Jillson, P. Saldin, K. Stuopis, Q. Wang, and K. Xue