Hyperbolic lines under the upper half-plane model are represented in two ways (see Figure 1):

1.       as rays emanating from points on the x-axis and perpendicular to the x-axis.
2.       as semi-circles in the upper half-plane whose center lies on the x-axis.

Figure 1

Given:

Two points P1 and P2 in the upper half-plane.

To Construct:

The hyperbolic line (u-line) that passes through the two given points.

Constructions:

Figure 2

1.   Let m be the perpendicular bisector of P1 and P2, and let C be the intersection of m and the x-axis (see Figure 2).
   
   
2.       Let Q1 and Q2 be intersections of the x-axis and the circle with center C and radius , and then the final object (u-line) is the arc (semi-circle) _.

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