looking for sanity checks in an insane world

Some years ago, using basic theorems of hyperbolic trigonometry, I worked out a conformal representation of the hyperbolic plane which preserves one line – analogous to the Mercator projection of the sphere, a conformal map which preserves one great circle. (At that time, I had a mistaken notion of the Mercator rule, so I didn’t know until later how strong the analogy is.) Unfortunately I’ve utterly forgotten how I worked it out; and I need to try again; because, now that I’ve learned enough hyperbolic analytic geometry to try making some pictures, there appears to be a flaw in my formula.