Start with a u-line m with end points A and B, and a point P not on m in the upper half-plane (see Figure 10). Consider the locus of all points on the same side of m as P and at the same perpendicular distance from m as P in non-Euclidean sense. This locus is called hypercycle or equidistant curve, and is represented by the arc of the Euclidean circle passing through A, B, and P.

Figure 10

Given:

A u-line m with boundary A, B and a point P not on m in the upper half-plane.

To Construct:

The hypercycle (equidistant curve) passing through P and equidistant to m.

Constructions:

According to the above statement, we can obtain the required hypercycle by simply joining the arc .

Download This Macro hypercycle.mac