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There
are two types of parallel lines in hyperbolic geometry (see Figure
16). Neither n nor q has common point with the third u-line m
in the upper half-plane. Both n and q are parallel u-lines to
m, but there is a difference between them. U-line n has common
end (ideal point) with m at E, but q does not. We say that u-lines
q and m are ultra parallel, and u-lines n and m are limiting
parallel.
 Figure 16
Given:
A u-line m with a point P not on it in the
upper half-plane.
To Construct:
Two limiting parallel u-lines of m through
P.
Constructions:
 Figure 17
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Let A and B be ends of m and
let C be the intersection of the x-axis and the perpendicular
bisector of
 .
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Let A' be symmetry of A with respect to C.
Then the arc
 is the
limiting parallel u-line of m through P with common boundary
A.
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The construction of the other
limiting parallel u-line through P with end B is similar.
 Download
This Macro two_limiting_parallel_u_lines.mac
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