Limiting parallel

In neutral or Absolute geometry, and in hyperbolic geometry there may be many lines parallel to a given line l through a point P not on line R; however, two parallels may be closer to l than all others. (one in each direction of 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.

Limiting parallels may sometimes form two, or three sides of a limit triangle.

Definition

The ray Aa is a limiting parallel to Bb, written: Aa|||Bb

A rays Aa is a limiting parallel to a ray Bb if they are coterminal or if they lie on distinct lines not equal to the line AB, they do not meet, and every ray in the interior of the angle BAa meets the ray Bb.[1]

Properties

Distinct lines carrying limiting parallel rays do not meet.

Proof

Suppose that the lines carrying distinct parallel rays met. By definition the cannot meet on the side of AB which either a is on. Then they must meet on the side of AB opposite to a, call this point C. Thus \angle CAB + \angle CBA < 2 \text{ right angles} \Rightarrow \angle aAB + \angle bBA > 2 \text{ right angles} . Contradiction.


See Also

References

  1. ↑ Hartshorne, Robin (2000). Geometry: Euclid and beyond (Corr. 2nd print. ed.). New York, NY [u.a.]: Springer. ISBN 978-0-387-98650-0.