In hyperbolic geometry, the Ultraparallel theorem states that every pair of ultraparallel lines in the hyperbolic plane has a unique common perpendicular hyperbolic line. A triangle immersed in a saddle-shape plane, as well as two diverging parallel lines. ... A perpendicular 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. Abscissa means the x coordinate on an (x, y) graph; the input of a mathematical function against which the output is plotted. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In physics, hyperbolic motion is the motion of an object with constant acceleration in special relativity. ...

Compose the following hyperbolic motions:

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Then , , , .

Then a stays at , , , (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 . Since is the radius of the semicircle on 1z, the common perpendicular sought has radius-square

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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 to yield the unique hyperbolic line perpendicular to both ultraparallels p and q.