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Ultraparallel theorem
 
 
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In hyperbolic geometry
Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P th...
, the ultraparallel theorem states that every pair of ultraparallel lines in the hyperbolic plane has a unique common perpendicular
Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
 hyperbolic line.

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be four distinct points on the abscissa of the Cartesian plane. Let and be semicircles above the abscissa with diameters and respectively. Then in the Poincaré half-plane model
Poincaré half-plane model

In non-Euclidean geometry, the Poincar? half-plane model is the upper half-plane, together with a metric, the Poincar? metric, that makes it a model of two-dimensional hyperbolic geometry....
 HP, and represent ultraparallel lines.

Compose the following two hyperbolic motion
Hyperbolic motion

In geometry, a hyperbolic motion is a mapping of a model of hyperbolic geometry that preserves the distance measure in the model. Such a mapping is analogous to congruences of Euclidean geometry which are compositions of rotations and translations....
s:

Then ,

Now continue with these two hyperbolic motions:

Then stays at , , , (say).






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In hyperbolic geometry
Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P th...
, the ultraparallel theorem states that every pair of ultraparallel lines in the hyperbolic plane has a unique common perpendicular
Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
 hyperbolic line.

Proof in the Poincaré half-plane model


Let

be four distinct points on the abscissa of the Cartesian plane. Let and be semicircles above the abscissa with diameters and respectively. Then in the Poincaré half-plane model
Poincaré half-plane model

In non-Euclidean geometry, the Poincar? half-plane model is the upper half-plane, together with a metric, the Poincar? metric, that makes it a model of two-dimensional hyperbolic geometry....
 HP, and represent ultraparallel lines.

Compose the following two hyperbolic motion
Hyperbolic motion

In geometry, a hyperbolic motion is a mapping of a model of hyperbolic geometry that preserves the distance measure in the model. Such a mapping is analogous to congruences of Euclidean geometry which are compositions of rotations and translations....
s:

Then ,

Now continue with these two hyperbolic motions:

Then stays at , , , (say). The unique semicircle, with center at the origin, perpendicular to the one on must have a radius tangent to the radius of the other. The right triangle formed by the abscissa and the perpendicular radii has hypotenuse of length . Since is the radius of the semicircle on , the common perpendicular sought has radius-square

The four hyperbolic motions that produced above can each be inverted and applied in reverse order to the semicircle centered at the origin and of radius to yield the unique hyperbolic line perpendicular to both ultraparallels and .

Proof in the Klein model

In the Klein model
Klein model

In geometry, the Klein model, also called the projective model, the Beltrami?Klein model, the Klein?Beltrami model and the Cayley?Klein model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or ball, and the lines of the geometry are line segments contained in the disk; that i...
 of the hyperbolic plane, two ultraparallel lines correspond to two non-intersecting chords. The poles
Pole and polar

In geometry, the terms pole and polar are used to describe a point and a line that have a unique reciprocal relationship with respect to a given conic section....
 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 whose extension passes through the pole of A. 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 plane, and so we draw a line through the pole of the other line intersecting the diameter at right angles to get the common perpendicular.

The proof is completed by showing this construction is always possible. If both chords are diameters, they intersect. If only one of the chords is a diameter, the 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.