  In previous work, we developed tools for quantifying the tidal distortion of
a black hole's event horizon due to an orbiting companion. These tools use
techniques which require large mass ratios (companion mass $\mu$ much smaller
than black hole mass $M$), but can be used for arbitrary bound orbits, and for
any black hole spin. We also showed how to visualize these distorted black
holes by embedding their horizons in a global Euclidean 3-space,
${\mathbb{E}}^3$. Such visualizations illustrate interesting and important
information about horizon dynamics. Unfortunately, we could not visualize black
holes with spin parameter $a_* > \sqrt{3}/2 \approx 0.866$: such holes cannot
be globally embedded into ${\mathbb{E}}^3$. In this paper, we overcome this
difficulty by showing how to embed the horizons of tidally distorted Kerr black
holes in a hyperbolic 3-space, ${\mathbb{H}}^3$. We use black hole perturbation
theory to compute the Gaussian curvatures of tidally distorted event horizons,
from which we build a two-dimensional metric of their distorted horizons. We
develop a numerical method for embedding the tidally distorted horizons in
${\mathbb{H}}^3$. As an application, we give a sequence of embeddings into
${\mathbb{H}}^3$ of a tidally interacting black hole with spin $a_*=0.9999$. A
small amplitude, high frequency oscillation seen in previous work shows up
particularly clearly in these embeddings.
