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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.
Compose 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 of length 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.
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