Ultraparallel theorem: Difference between revisions

Revision as of 23:45, 10 August 2019

Revision as of 14:04, 17 December 2020

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If not, we may suppose AB < CB' without loss of generality. Let E be a point on the line s on the opposite side of A from C. Take A' on CB' so that A'B' = AB. Through A' draw a line s' (A'E') on the side closer to E, so that the angle B'A'E' is the same as angle BAE. Then s' meets s in an ordinary point D'. Construct a point D on ray AE so that AD = A'D'.

Then D' ≠ D. They are the same distance from r and both lie on s. So the perpendicular bisector of D'D (a segment of s) is also perpendicular to r.<ref>{{cite book|last1=[[H. S. M. Coxeter]]|author1-link=H. S. M. Coxeter|title=Non-euclidean Geometry|isbn=978-0-88385-522-5|pages=190-192190–192}}</ref>

(If r and s were asymptotically parallel rather than ultraparallel, this construction would fail because s' would not meet s. Rather s' would be asymptotically parallel to both s and r.)

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Let

:<math>a < b < c < d</math>

be four distinct points on the [[abscissa]] of the [[Cartesian plane]]. Let <math>p</math> and <math>q</math> be [[semicircle]]s above the abscissa with diameters <math>ab</math> and <math>cd</math> respectively. Then in the [[Poincaré half-plane model]] HP, <math>p</math> and <math>q</math> represent ultraparallel lines.

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* The [[Pole and polar|poles]] of these two lines are the respective intersections of the [[tangent line]]s to the boundary [[circle]] at the endpoints of the chords.

* Lines ''perpendicular'' to line ''l'' are modeled by chords whose extension passes through the pole of ''l''.

* Hence we draw the unique line between the poles of the two given lines, and intersect it with the boundary circle ; 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 it is also perpendicular in the Beltrami-Klein model, and so we draw a line through the pole of the other line intersecting the diameter at right angles to get the common perpendicular.

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==References==

{{reflistReflist}}

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