A Bryant surface (or catenoid cousin) is a two-dimensional surface embedded in three-dimensional hyperbolic space with constant mean curvature equal to 1. Bryant derived a holomorphic parameterization for such surfaces, similar to the Weierstrass–Enneper parameterization for minimal surfaces [1, 2].
A one-parameter family of these surfaces of revolution is defined for the parameter ; for the surface is embedded, and for the surface is not embedded; as tends to zero, the surfaces converge to two horospheres [3, 4, 5]. Another family that arises consists of warped surfaces that are not surfaces of revolution with two regular embedded ends, with warps and parameters and [6, 7]. The parameters and control the range over which the surface is plotted.