Under the upper half-plane model, let m be a u-line and let n be another u-line perpendicular to m at Q with boundary S (see Figure 8). Let r be a u-circle passing through Q and with its hyperbolic center P on n. Now let P recede from Q along the perpendicular u-line n towards the ideal point S. The u-circle r will approach a limiting position h as P recedes arbitrarily far from Q. The limiting position of r is called a horocycle or limiting curve. It is the Euclidean circle passing through the ideal point S and tangent to the x-axis (the dotted circle as shown).

Figure 8

Given:

A point P1 lying on the x-axis and another point P2 in the upper half-plane.

To Construct:

The horocycle (limiting curve) that passes through P1 and P2.

Constructions:

Figure 9

1.      First, construct the perpendicular bisector n of points P1 and P2.
   
   
2.       Let  be the straight-line, which is perpendicular to the x-axis at P1 and intersects n at C.
   
   
3.       The Euclidean circle determined by center C and radius  is then the required horocycle h.

Download This Macro horocycle.mac