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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?

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  • $\begingroup$ In a word, yes. $\endgroup$ – Lord Shark the Unknown Jul 1 '18 at 19:46
  • $\begingroup$ 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. $\endgroup$ – mweiss Jul 1 '18 at 20:06
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    $\begingroup$ 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). $\endgroup$ – Rocket Man Jul 1 '18 at 21:36
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Terminology does vary a little, so I won't use the word parallel. I do not know the book you mention.

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 $\pi.$

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