Horosphere in hyperbolic three-space

A horosphere in hyperbolic three-space is an isometric image of a horizontal plane $r=r_0>0$ in the upper-half-space model

\mathbb H^3=\{(z,r):z\in\mathbb C,\ r>0\},\qquad ds^2=(|dz|^2+dr^2)/r^2.

The point $\infty$ is the ideal center of a horizontal horosphere; isometries transport it to the center of the image horosphere.

Other centers

Horospheres centered at a finite boundary point are Euclidean spheres tangent to $r=0$ , with the tangency point omitted. The induced metric on a horizontal horosphere is $|dz|^2/r_0^2$ , so it is intrinsically Euclidean.