Ultraparallel theorem: Difference between revisions

Revision as of 19:24, 19 December 2020

Revision as of 19:25, 19 December 2020

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<!-- ??? "then we may extend the tangents drawn from each pole to produce a [[quadrilateral]] with the unit circle inscribed within it " this is not always the case , they not always form a quadrilateral, nor is the quadrilateral always convex see also http://math.stackexchange.com/q/1382739/88985 -->

Alternatively, we can construct the common perpendicular of the [[ultraparallel]] lines as follows: the ultraparallel lines in Beltrami-Klein model are two non-intersecting chords. But they actually intersect outside the circle. The polar of the intersecting point is the desired common perpendicular.<ref>W. Thurston, ''Three-Dimensional Geometry and Topology'', page 72</ref>

==References==