I've recently learned about [Amsler's surface](http://resolver.sub.uni-goettingen.de/purl?GDZPPN002284723), a surface of constant negative Gaussian curvature. If I understand things correctly, there is a whole family of such surfaces, differing in the angle of intersection for the two lines that generate it. But I guess I'm only interested in the most symmetric version, where these lines are orthogonal. I'd like to know how large a portion of the hyperbolic plane I can embed on this surface.

So if the unit of length is chosen such that the Gaussian curvature of the surface becomes $-1$, **what is the radius of a circle which** is centered at the intersection of the asymptotic lines and **just touches the cuspidal edges** of the surface? A “circle” here would be the set of points on the surface with fixed geodesic distance to a central point. That intrinsic circle would not be a planar circle in the 3D embedding of the surface.

The following illustration of Amsler's surface was taken from the [*Gallery of pseudospherical surfaces*](http://www.cns.gatech.edu/~danek/preprints/ovchinnikov1.ps.gz) by A. Ovchinninkov. I'm not sure how accurately it matches what I ask for, since there are [other figures on the web](http://www.google.com/search?q=amsler%20surface&tbm=isch) which look somewhat different from this.

![Fig. 14 of the referenced document](https://i.stack.imgur.com/bs8uL.png)