Hyperbolic geometry
 parallel lines

Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry:

This implies that there are through P an infinite number of coplanar lines that do not intersect R.

These non-intersecting lines are divided into two classes:

Some geometers simply use the phrase " parallel lines" to mean " limiting parallel lines", with ultraparallel lines meaning just non-intersecting.

These limiting parallels make an angle θ with PB; this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism.

For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane that is perpendicular to each pair of ultraparallel lines.
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