Helicoid in H^3

Here are three views of a hyperbolic helicoid, an idea suggested by +Saul Schleimer. The first puts the helicoid axis on the vertical line through the origin. The other two apply a hyperbolic isometry to move the axis towards the right. Whenever the axis does not go through the origin, it will be an arc in the ball model.

What curve does a single ideal edge of the surface trace out on the ball boundary? I suspect it is a loxodrome, but I haven't proved this. (Update: It is! see comments below.)

Relevant Links

Euclidean helicoids
en.wikipedia.org/wiki/Helicoid

Loxodromes
en.wikipedia.org/wiki/Rhumb_line
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H3 Ruled Surfaces
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