Ultraparallel theorem: Difference between revisions

Revision as of 23:14, 20 February 2008

Revision as of 02:56, 2 March 2008

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The four hyperbolic motions that produced <math>z</math> above can each be inverted and applied in reverse order to the semicircle centered at the origin and of radius <math>\sqrt{z}</math> to yield the unique hyperbolic line perpendicular to both ultraparallels <math>p</math> and <math>q</math>.

==Proof 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 modeled by chords such that when extended, thewhose 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 lines. 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 spaceplane, and so we draw a line through the pole of the other line intersecting the diameter at right angles to get the ultraparallelcommon lineperpendicular.

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

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other chord projects orthogonally down to a section of the first chord contained in its interior, and a line from the pole orthogonal to the diameter intersects both the diameter and the chord. If both lines are not diameters, the we may extend the tangents drawn from each pole to produce a quadrilateral with the unit circle inscribed within it. The poles are opposite vertices of this quadrilateral,

and the chords are lines drawn between adjacent sides of the vertex, across opposite corners. Since the quadrilateral is convex, the line between the poles intersects both of the chords drawn across the corners, and the segment of the line between the chords defines the required chord perpendicular to the two other chords.

==References==

* [[Karol Borsuk]] & Wanda Szmielew (1960) ''Foundations of Geometry'', page 291.