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 a < b < c < d be four distinct points on the abscissa of the Cartesian plane. Let p and q be semicircles above the abscissa with diameters ab and cd respectively. Then in the upper half-plane model HP , p and q represent ultraparallel lines. Consider the following hyperbolic motions:
- x → x – a
- inversion in the unit semicircle.
Then a → ∞ , b → (b – a)-1, c → (c – a)-1 , d → (d – a)-1.
- x → x – (b – a)-1
- x → [(c-a)-1 - (b-a)-1]-1 x
Then a stays at ∞, b → 0 , c → 1 , d → z (say). The unique semicircle, with center at the origin, perpendicular to the one on 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 b = ½(z + 1).Since ½(z – 1) is the radius of the semicircle on 1z, the common perpendicular sought has radius-square
- ¼[(z + 1)2 - (z – 1)2] = z .
The four hyperbolic motions that produced z above can each be inverted and applied in reverse order to the semicircle centered at the origin and of radius √z to yield the unique hyperbolic line perpendicular to both ultraparallels p and q.