Is there a compass and straightedge construction of parallel lines in hyperbolic geometry? That is, given a line and a point not on the line, construct a line parallel to the given line.

$\begingroup$ what do you mean by "parallel"? "not intersecting"? Then take your point A, line l, draw perpendicular AB to l, then perpendicular m to AB through A. $\endgroup$ – Fedor Petrov Oct 25 '10 at 18:01

1$\begingroup$ I think this depends on what model you are using. For example, you can do this in the Poincare disk model of the hyperbolic plane. Or maybe you mean you have a hyperbolic compass (i.e. given a point and a radius, one can construct a hyperbolic circle centered at that point) and a hyperbolic ruler (i.e. given two distinct points, one can construct the geodesic line thru them). $\endgroup$ – Yi Liu Oct 25 '10 at 18:03

2$\begingroup$ No I am talking about one of the two limiting lines that are asymptotic not intersecting; called parallel lines in Coxeter. Lines not intersecting that have sharing a common perpendicular are called ultraparallel by Coxeter. In everthing I have looke at I have seen no construction by a compass and straght edge. $\endgroup$ – dlb Oct 25 '10 at 18:14
The quickest way to get you started is to refer you to my article, reference [5] (a pdf) on
http://en.wikipedia.org/wiki/Squaring_the_circle
and then to the fourth edition (2008) of Marvin Jay Greenberg's book, which is reference [6].
I'm guessing what you want is Bolyai's construction, given a line and a point off the line, of the two rays through the point that are asymptotic to the line, one in each direction. When I wrote the article, I relied on an earlier edition of Marvin's book, along with The Foundations of Geometry and the NonEuclidean Plane by George E. Martin, which has a nice little section at the very end. There is also, now, Geometry: Euclid and Beyond by Robin Hartshorne.
The most complete reference I know on constructions is in Russian, by Smogorshevski, other very helpful books by Kagan and by Nestorovich. Of course, at this point I have my own versions of it all.

1$\begingroup$ Perhaps, you are thinking of this little book which was in fact translated: A.S. Smogorzhevskij, The ruler in geometrical construction, Pergamon Press, Oxford, 1961. $\endgroup$ – Igor Pak Oct 27 '10 at 0:08

$\begingroup$ Hi, Igor. No, it is the same guy but 1951, "Geometric Constructions in the Lobachevskii Plane" Math Reviews 14, 575. It is a reference in my little paper and in one by Robert R. Curtis, approximately 1990, I do not seem to have the title for Curtis. I photocopied all the Russian and a few Ukrainian items, MordukhaiBoltovskoi wrote in many languages. Then my friend Dmitry would read me translations of a page or two by telephone. I have a complete copy of Mr. Smog around here somewhere, I can't seem to find it. $\endgroup$ – Will Jagy Oct 27 '10 at 17:54

1$\begingroup$ Do you have a simple argument (i.e. with no calculations) showing that Bolyai's construction really does what it suppose to? $\endgroup$ – Anton Petrunin Oct 30 '10 at 4:13

$\begingroup$ Anton there is a proof in Coxeter's "NonEuclidean Geometry" (just got my copy a few days ago). The proof uses projective geometry. $\endgroup$ – dlb Nov 2 '10 at 14:25
So you mean you want to see a compass and straight edge construction in Euclidean geometry of a circle passing through 2 given points and perpendicular to the given circle (which contains one of the points)? I believe that this construction is given in the geometry book by Robin Hartshorne, excellent book by the way.