Ultraparallel theorem

In hyperbolic geometry, the ultraparallel theorem states that every pair of ultraparallel lines in the hyperbolic plane has a unique common perpendicular hyperbolic line.

Let

${\displaystyle a<b<c<d}$

be four distinct points on the abscissa of the Cartesian plane. Let ${\displaystyle p}$ and ${\displaystyle q}$ be semicircles above the abscissa with diameters ${\displaystyle ab}$ and ${\displaystyle cd}$ respectively. Then in the upper half-plane model HP, ${\displaystyle p}$ and ${\displaystyle q}$ represent ultraparallel lines.

Compose the following two hyperbolic motions:

${\displaystyle x\to x-a\,}$

${\displaystyle {\mbox{inversion in the unit semicircle}}\,}$.

Then ${\displaystyle a\to \infty }$, ${\displaystyle b\to (b-a)^{-1}}$, ${\displaystyle c\to (c-a)^{-1}}$, ${\displaystyle d\to (d-a)^{-1}}$.

Now continue with these two hyperbolic motions:

${\displaystyle x\to x-(b-a)^{-1}\,}$

${\displaystyle x\to \left[(c-a)^{-1}-(b-a)^{-1}\right]^{-1}x\,}$

Then ${\displaystyle a}$ stays at ${\displaystyle \infty }$, ${\displaystyle b\to 0}$, ${\displaystyle c\to 1}$, ${\displaystyle d\to z}$ (say). The unique semicircle, with center at the origin, perpendicular to the one on ${\displaystyle 1z}$ 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 ${\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}(z+1)}$. Since ${\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}(z-1)}$ is the radius of the semicircle on ${\displaystyle 1z}$, the common perpendicular sought has radius-square

${\displaystyle {\frac {1}{4}}\left[(z+1)^{2}-(z-1)^{2}\right]=z\,}$.

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