Section 5.3 Angle of Parallelism
¶Given a hyperbolic line \(l\) and a point \(P\) not on \(l\text{,}\) we can explore the lines through \(P\) that are, and are not, parallel to \(l\text{.}\) One possibility, illustrated in Figure 5.3.1, is to start by constructing the line through \(P\) that is perpendicular to \(l\text{.}\) All other lines through \(P\) will make some angle with this perpendicular line, and we can ask how that angle varies.
In Figure 5.3.1, parallel lines through \(P\) are represented by the line segment from \(B\) to \(P\text{,}\) and non-parallel lines are represented by the segment from \(A\) to \(P\text{.}\) In both cases, the corresponding angles are shown.
Activity 5.3.1. Exploring the angles created by parallel lines.
How do the angles shown in Figure 5.3.1 change as these points are moved? What happens if \(B\) approaches \(A\text{?}\) What happens if either or both points approach the boundary? (Warning: moving \(A\) also moves the line!)
Recall that the boundary points on the unit circle are not in the Poincaré disk. Thus, the intersection of \(l\) with the unit circle is not a point in the model. In the limiting case as \(A\) or \(B\) reaches this point, the resulting line is parallel to \(l\text{!}\) The corresponding angle is called the angle of parallelism \(\theta_0\) for \(P\) and \(l\text{.}\) This line is the “first” parallel line; if the angle is less than \(\theta_0\text{,}\) the line must intersect \(l\text{,}\) whereas if the angle is greater than or equal to \(\theta_0\text{,}\) the line is parallel to \(l\text{.}\)
Activity 5.3.2.
The angle of parallelism \(\theta_0\) is illustrated in Figure 5.3.2. What happens to \(\theta_0\) as the points are moved around? What if \(P\) approaches \(l\text{?}\) What if \(P\) approaches the boundary?
It should be readily apparent that \(\theta_0\) decreases as \(P\) approaches the boundary. Less obvious is that \(\theta_0\) depends only on the (hyperbolic) distance from \(P\) to \(l\text{,}\) measured of course along the perpendicular line, as illustrated in Figure 5.3.3. This result also explains why the angle of parallelism is the same on both sides, as shown in Figure 5.3.2.
The intuitive justification for this dependence of the angle of parallelism on distance involves applying SAS congruence to so-called ideal triangles that involve ideal points on the boundary “at infinity”. Formal proofs avoid these complications by applying SAS only to finite triangles, in most cases using proof by contradiction. We omit the details.