0709 Constructing ultraparallel lines
Problem:
In the hyperbolic plane given line a and point P outside the line. Construct the two lines
through P that are asymptotically parallel to a.
Solution:
We follow János Bolyai's brilliant construction (see 34.§, pp. 107-108, 165-166):
- Let T be the foot of perpendicular from P to a, and let A be an arbitrary point of a that differs from T.
- Construct the quadrilateral PTAF such that the angles at vertices T, A and F are right angles.
- Let H be a point of segment PT such that AH=FP.
- Construct each angle with vertex P that are congruent to α = ∠AHT so that one side of the angle is PT. The other side of this angle (that differs from PT) will be asymptotically parallel to a.
070901
Discussion:
The obtained angle has two sides: the perpendicular from P to a, and the line through P that is
asymptotically parallel to a. Bolyai called this angle the angle of snapping (today the angle of parallelism is used in English), which is a very suggestive definition. He also found an important relationship between the angle of snapping and the length of the segment PT, which is called distance of parallelism.
In mathematics, “showing” some property is usually a synonym of “proving” something. But in our case we will prefer the meaning “visualizing” when doing experiments in the P-model. For instance, by using the means of the P-model it can be shown that the construction is accurate within 10-8 mm.
We can find further important relationships in this construction. Let A0 be the intersection of a circle with center P and circumpoint F, and draw the line through P which is asymptotically parallel to [T,A). Furthermore let T0 be the perpendicular projection of point A0 on the line (PT). It can be shown that:
- rays [T0A0) and [PF) are asymptotically parallel to each other,
- points A0 and A are symmetric w.r.t. the perpendicular bisector of segment T0T.
We constructed the quadrilateral PTAF with three right angles. It is called a Lambert quadrilateral. The Swiss mathematician J. H. Lambert (1728-1777) did remarkable work on computing the measure of the fourth angle. In Euclidean geometry this angle is also a right angle, while in spherical geometry it is greater than 90°. Lambert considered the option that the fourth angle was less than a right angle, but—unlike Bolyai and Lobachevsky—he eventually rejected it because of the bizarre and contradictory appearance of this quadrilateral from the Euclidean point of view.
Hopefully the reader holds a different opinion after studying the P-model!