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Next: Hyperparallel Lines Up: Classification of Parallels Previous: Fan Angles

Limiting Parallel Rays

 

Consider a ray tex2html_wrap_inline12640 and a point tex2html_wrap_inline16466. Characterize all rays tex2html_wrap_inline15776 satisfying:

  1. tex2html_wrap_inline16470,
  2. every ray between tex2html_wrap_inline15776 and tex2html_wrap_inline16474 intersects tex2html_wrap_inline12640.
In euclidean geometry, there is only one such ray. Is this the case in tex2html_wrap_inline15734?

There is a fan angle of P and tex2html_wrap_inline12754, call it tex2html_wrap_inline16484, and one side of this angle lies on the same side of tex2html_wrap_inline14614 as B. Let us say that it is tex2html_wrap_inline15776. Now, tex2html_wrap_inline16470. Also, if tex2html_wrap_inline15840 lies between tex2html_wrap_inline15776 and tex2html_wrap_inline16474, then tex2html_wrap_inline16500 subdivides angle XPY. Thus, tex2html_wrap_inline16504, but due to its relative position with respect to tex2html_wrap_inline16474 and tex2html_wrap_inline15776 it must, in fact, intersect tex2html_wrap_inline12640. Thus, the side of the fan angle satisfies the two aforementioned conditions. Do any others?

Let tex2html_wrap_inline16296 be different from tex2html_wrap_inline15776.

If tex2html_wrap_inline16296 is on the line tex2html_wrap_inline14630, then tex2html_wrap_inline16520 intersects tex2html_wrap_inline12754 and the ray tex2html_wrap_inline16296 does not have property (1).
If tex2html_wrap_inline16296 lies on the same side of tex2html_wrap_inline14630 as Y, let tex2html_wrap_inline16532 be between tex2html_wrap_inline16474 and tex2html_wrap_inline16296. Then the line tex2html_wrap_inline14630 separates tex2html_wrap_inline16296 and tex2html_wrap_inline16532 from tex2html_wrap_inline12640, which means that tex2html_wrap_inline16546, so that tex2html_wrap_inline16296 does not satisfy property (2).
If tex2html_wrap_inline16296 lies on the same side of tex2html_wrap_inline14630 as X, then one of tex2html_wrap_inline16296 and tex2html_wrap_inline15776 is between the other and tex2html_wrap_inline16474.
(i)
If tex2html_wrap_inline16296 is between tex2html_wrap_inline15776 and tex2html_wrap_inline16474 then tex2html_wrap_inline16296 must intersect tex2html_wrap_inline12754 by the properties of the fan angle. Thus, tex2html_wrap_inline16296 would not satisfy property (1).
(ii)
If tex2html_wrap_inline15776 is between tex2html_wrap_inline16474 and tex2html_wrap_inline16296 then tex2html_wrap_inline16296 cannot satisfy property (2), for tex2html_wrap_inline16582.

Thus, tex2html_wrap_inline15776 is the only ray at P with these two properties. This gives rise to the following definition.

Definition: A ray tex2html_wrap_inline13456 is parallel to a ray tex2html_wrap_inline12640 if

  1. tex2html_wrap_inline16592,
  2. every ray between tex2html_wrap_inline13456 and tex2html_wrap_inline16474 intersects tex2html_wrap_inline12640.

We have just shown that this ray is unique. In this case the rays tex2html_wrap_inline13456 and tex2html_wrap_inline12640 are called limiting parallel rays and we say that tex2html_wrap_inline13456 is limiting parallel to tex2html_wrap_inline12640. 

QUESTION: If tex2html_wrap_inline13456 is limiting parallel to tex2html_wrap_inline12640, is tex2html_wrap_inline12640 is limiting parallel to tex2html_wrap_inline13456?

We need to following lemma for the proof of the main theorem. I leave the proof to you as an exercise.

Lemma 13.1:  If tex2html_wrap_inline16616 is acute, then the foot of P in the line tex2html_wrap_inline12754 lies in the ray tex2html_wrap_inline12640 and is different from A.

I offer the following theorems without proof.

Theorem 13.2:  If tex2html_wrap_inline13456 is limiting parallel to tex2html_wrap_inline12640, then tex2html_wrap_inline12640 lies in the interior of the angle tex2html_wrap_inline15624.

Theorem 13.3:  Let tex2html_wrap_inline13682 and let tex2html_wrap_inline12498 in tex2html_wrap_inline11154. Then tex2html_wrap_inline13456 is limiting parallel to tex2html_wrap_inline12640 if and only if tex2html_wrap_inline13456 is limiting parallel to tex2html_wrap_inline13316.

Theorem 13.4:  Let tex2html_wrap_inline12498 on tex2html_wrap_inline11154. tex2html_wrap_inline12640 is limiting parallel to tex2html_wrap_inline16654 if and only if tex2html_wrap_inline13316 is limiting parallel to tex2html_wrap_inline16654.

Theorem 13.5: If tex2html_wrap_inline13456 is limiting parallel to tex2html_wrap_inline12640, then tex2html_wrap_inline12640 is limiting parallel to tex2html_wrap_inline13456.

figure3767

Proof: Let F be the foot of P in tex2html_wrap_inline16694. Let tex2html_wrap_inline13256 so that tex2html_wrap_inline16698 is like directed to tex2html_wrap_inline12640. This means that either tex2html_wrap_inline16702 or tex2html_wrap_inline16704. It follows from Theorem 14.5 that tex2html_wrap_inline13456 is limiting parallel to tex2html_wrap_inline16698, so that C lies on the same side of tex2html_wrap_inline16712 as does Q and tex2html_wrap_inline16716. Let tex2html_wrap_inline16718 lie in the interior of tex2html_wrap_inline16720.

CLAIM: tex2html_wrap_inline16722.

From this it follows that tex2html_wrap_inline16698 is limiting parallel to tex2html_wrap_inline13456. Applying Theorem 14.5 again, we have that tex2html_wrap_inline12640 is limiting parallel to tex2html_wrap_inline13456. Thus, we need to establish this claim to finish the proof.

Since tex2html_wrap_inline16732, tex2html_wrap_inline16734 is acute. Thus, by Lemma 14.1 the foot of P in tex2html_wrap_inline16738 must lie in the ray tex2html_wrap_inline16718. Label this point G. It follows that G lies on the same side of tex2html_wrap_inline16712 as C.

If tex2html_wrap_inline16750, then it must lie in tex2html_wrap_inline13456. It follows that tex2html_wrap_inline16722, and we are done.

If G lies on the opposite side of tex2html_wrap_inline15614 from F, then again tex2html_wrap_inline16718 intersects tex2html_wrap_inline13456 and we are done.

Assume then, as the final case, that G lies on the same side of tex2html_wrap_inline15614 as F. Then, this puts G in the interior of tex2html_wrap_inline16774. Let tex2html_wrap_inline16776. We have that tex2html_wrap_inline16778. By Congruence Axiom 4 there is a unique ray tex2html_wrap_inline16780 in the interior of tex2html_wrap_inline16774 so that
displaymath16460
Since tex2html_wrap_inline13456 is limiting parallel to tex2html_wrap_inline16698, tex2html_wrap_inline16780 must intersect tex2html_wrap_inline16698 in some point D. From Lemma 14.1 tex2html_wrap_inline16794, so that PG<PF by an Exercise (the hypotenuse of a right triangle is longer than either leg). Thus, there is a point tex2html_wrap_inline16798 so that tex2html_wrap_inline16800.

Let tex2html_wrap_inline16802 be perpendicular to tex2html_wrap_inline16712 at H. Since tex2html_wrap_inline12754 is perpendicular to tex2html_wrap_inline16712 it follows that tex2html_wrap_inline16812. Now, tex2html_wrap_inline16802 does intersect one side of the triangle tex2html_wrap_inline16816. By Pasch's Theorem it must intersect a second side. That side must be PD. Let the point of intersection be E. Now, tex2html_wrap_inline16822 and tex2html_wrap_inline16824.

On tex2html_wrap_inline13456 there is a unique point M so that tex2html_wrap_inline16830. Recalling that tex2html_wrap_inline16800 and tex2html_wrap_inline16834, we then have that tex2html_wrap_inline16836 by SAS. Thus, tex2html_wrap_inline16838 and is a right angle. Thus, tex2html_wrap_inline16840 since both are perpendicular to tex2html_wrap_inline16842 at G. Therefore, tex2html_wrap_inline16846. Since M lies on the same side of tex2html_wrap_inline16850 as Q, it follows, finally, that tex2html_wrap_inline16854 and tex2html_wrap_inline16722.

A ray tex2html_wrap_inline16858 is parallel to a line tex2html_wrap_inline11154 if tex2html_wrap_inline16862 is a limiting parallel ray to some ray in tex2html_wrap_inline11154. A line k is limiting parallel  or asymptotically parallel , or even horoparallel , to a line tex2html_wrap_inline11154 if some ray in k is a limiting parallel ray to some ray in tex2html_wrap_inline11154. We have just proven that these parallelisms are symmetric and we may denote them by
displaymath16461

If tex2html_wrap_inline15470, tex2html_wrap_inline15472, and tex2html_wrap_inline16878, then tex2html_wrap_inline12640 is said to be parallel to tex2html_wrap_inline11154 in the direction of tex2html_wrap_inline14770 on tex2html_wrap_inline11154. Furthermore, k and tex2html_wrap_inline11154 are said to be parallel in the direction of tex2html_wrap_inline12640 on k and in the direction of tex2html_wrap_inline14770 on tex2html_wrap_inline11154.

Theorem 13.6:  If tex2html_wrap_inline13682 then there are exactly two lines through P that are limiting parallel to tex2html_wrap_inline11154. Each contains an arm of the fan angle tex2html_wrap_inline16434 and they are limiting parallel to tex2html_wrap_inline11154 in opposite directions.

As we have mentioned several times already, there is no simple transitivity of parallelism in hyperbolic geometry. There is a weak form of transitivity.

Theorem 13.7:[Weak Transitivity of Parallels] Two lines parallel to a third in the same direction on the third are parallel to each other. 

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Next: Hyperparallel Lines Up: Classification of Parallels Previous: Fan Angles

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