# Are circles touching at the real line considered parallel in the hyperbolic plane?

In James Anderson's Hyperbolic Geometry, two lines in the upper half-plane model of the hyperbolic plane are said to be parallel if they are disjoint.

Suppose $l_1$ is the half-circle centered at $(0, 0)$ with radius $1$, and $l_2$ is the half-circle centered at $(2, 0)$ with radius 1. These circles intersect at $(1, 0)$, but this point is not a part of the upper half-plane. Would it be right to say that $l_1$ and $l_2$ are parallel?

• In a word, yes. – Lord Shark the Unknown Jul 1 '18 at 19:46
• Yes. The same answer holds if one semicircle is internally tangent to another with their intersection on the x-axis: for example the semicircle centered at (0,0) with radius 1 and the semicircle centered at (-2,0) with radius 2. – mweiss Jul 1 '18 at 20:06
• If you like, you can make a distinction between geodesics that intersect in a common point of the boundary (asymptotically parallel) and those geodesics that don’t (ultra-parallel). – Rocket Man Jul 1 '18 at 21:36

One thing that is special about your lines is that there is no common perpendicular line to them. If you begin with a line segment and two lines orthogonal to it at both endpoints, we know that these lines cannot meet, as that would create a triangle with angle sum at least $\pi.$