Ultraparallel theorem: Difference between revisions

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In [[hyperbolic geometry]], thetwo non-intersecting lines that are ''not'' [[limiting parallel]] are called '''ultraparallel lines'''. The '''ultraparallel theorem''' states that every pair of [[ultraparallel line]]s (lines that are not intersecting and not [[limiting parallel]]) has a unique common [[perpendicular]] hyperbolic line.

==Hilbert's construction==

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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=coxeter[[H. S. M. Coxeter]]|title=non Non-euclidean geometryGeometry|isbn=978-0-88385-522-5|pages=190-192}}</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>

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:<math>\mbox{inversion in the unit semicircle.}</math>

Then <math>a \to \infty</math>, <math>\quad b \to (b-a)^{-1},\quad c \to (c-a)^{-1},\quad d \to (d-a)^{-1}.</math>

Now continue with these two hyperbolic motions:

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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'', p72page 72</ref>

==References==