In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l {\displaystyle l} through a point P {\displaystyle P} not on line R {\displaystyle R} ; however, in the plane, two parallels may be closer to l {\displaystyle l} than all others (one in each direction of R {\displaystyle R} ).
Thus it is useful to make a new definition concerning parallels in neutral geometry. If there are closest parallels to a given line they are known as the limiting parallel, asymptotic parallel or horoparallel (horo from Greek: ὅριον — border).
For rays, the relation of limiting parallel is an equivalence relation, which includes the equivalence relation of being coterminal.
If, in a hyperbolic triangle, the pairs of sides are limiting parallel, then the triangle is an ideal triangle.
Definition
A ray A a {\displaystyle Aa} is a limiting parallel to a ray B b {\displaystyle Bb} if they are coterminal or if they lie on distinct lines not equal to the line A B {\displaystyle AB} , they do not meet, and every ray in the interior of the angle B A a {\displaystyle BAa} meets the ray B b {\displaystyle Bb} .1
Properties
Distinct lines carrying limiting parallel rays do not meet.
Proof
Suppose that the lines carrying distinct parallel rays met. By definition they cannot meet on the side of A B {\displaystyle AB} which either a {\displaystyle a} is on. Then they must meet on the side of A B {\displaystyle AB} opposite to a {\displaystyle a} , call this point C {\displaystyle C} . Thus ∠ C A B + ∠ C B A < 2 right angles ⇒ ∠ a A B + ∠ b B A > 2 right angles {\displaystyle \angle CAB+\angle CBA<2{\text{ right angles}}\Rightarrow \angle aAB+\angle bBA>2{\text{ right angles}}} . Contradiction.
See also
- horocycle, In Hyperbolic geometry a curve whose normals are limiting parallels
- angle of parallelism
References
Hartshorne, Robin (2000). Geometry: Euclid and beyond (Corr. 2nd print. ed.). New York, NY [u.a.]: Springer. ISBN 978-0-387-98650-0. 978-0-387-98650-0 ↩