Limiting parallel

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

In neutral geometry, there may be many lines parallel to a given line l at a point P, however one parallel may be closer to l than all others. Thus it is useful to make a new definition concerning parallels in neutral geometry. If there is a closest parallel to a given line it is known as the limiting parallel. The relation of limiting parallel for rays 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[edit]

A ray 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[edit]

Distinct lines carrying limiting parallel rays do not meet.

Proof[edit]

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.

References[edit]

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