Ultraparallel theorem: Difference between revisions

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==Proof of the ultraparallel theorem in the Klein model==

In the [[Klein model]] of the hyperbolic plane, two ultraparallel lines correspond to two non-intersecting chords. The ''poles'' of these two lines are the respective intersections of the tangent lines to the unit circle at the endpoints of the chords. Lines perpendicular to line A are modelledmodeled by chords such that when extended, the extension passes through the pole of A, and vice-versa. Hence we draw the unique line between the poles of the two given lines, and intersect it with the unit disk; the chord of intersection will be the desired common perpendicular of the ultraparallel linelines. If one of the chords happens to be a diameter, we do not have a pole, but in this case any chord perpendicular to the diameter is perpendicular as well in the hyperbolic space, and so we draw a line through the pole of the other line intersecting the diameter at right angles to get the ultraparallel line.

The proof is completed by showing this construction is always possible. If both