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Is there a compass and straight edge construction of parallel lines in hyperbolic geometry?

That is given a line and a point not on the line construct one of the line parallel to the given line.

edited Apr 25 at 20:43

ε-δ
1,028●5●16

asked Oct 25 2010 at 17:44

dlb
136●4

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. – Fedor Petrov Oct 25 2010 at 18:01

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). – Yi Liu Oct 25 2010 at 18:03

1

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. – dlb Oct 25 2010 at 18:14

Will Jagy · Accepted Answer · 2010-10-25 18:43:55Z UTC

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 Non-Euclidean 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.

Beata Randrianantoanina · Answer 2 · 2010-11-19 12:57:44Z UTC

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.

Is there a compass and straight edge construction of parallel lines in hyperbolic geometry?

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