Angle of Parallelism
| Playfair's form of the Parallel Postulate says
that, for a fixed line and an intersecting line through a fixed point,
if we rotate the line about the fixed point then the point of intersection
moves along the line towards the infinite horizon. There is exactly one
line for which the intersecting point 'vanishes'. We can think of the lines
meeting on the horizon (at infinity).
In the Poincaré disc, if we move the intersecting line by grabbing the point M and rotating the line about the fixed point, the point of intersection again moves of towards the horizon in either direction, but now there are two lines that meet on the boundary of the disc, at points 1 and 2. Hence there are exactly two parallel lines. It we continue to rotate M about the fixed point we get the ultra parallel lines, those that neither intersect nor meet on the boundary |
A Line in the Poincaré discAngle of Parallelism
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