Is there a compass and straight edge construction of parallel lines in hyperbolic geometry? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:53:08Z http://mathoverflow.net/feeds/question/43554 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43554/is-there-a-compass-and-straight-edge-construction-of-parallel-lines-in-hyperbolic Is there a compass and straight edge construction of parallel lines in hyperbolic geometry? dlb 2010-10-25T17:44:49Z 2013-04-25T20:43:30Z <p>Is there a compass and straight edge construction of parallel lines in hyperbolic geometry?</p> <p>That is given a line and a point not on the line construct one of the line parallel to the given line. </p> http://mathoverflow.net/questions/43554/is-there-a-compass-and-straight-edge-construction-of-parallel-lines-in-hyperbolic/43558#43558 Answer by Will Jagy for Is there a compass and straight edge construction of parallel lines in hyperbolic geometry? Will Jagy 2010-10-25T18:43:55Z 2010-10-25T18:43:55Z <p>The quickest way to get you started is to refer you to my article, reference [5] (a pdf) on</p> <p><a href="http://en.wikipedia.org/wiki/Squaring_the_circle" rel="nofollow">http://en.wikipedia.org/wiki/Squaring_the_circle</a> </p> <p>and then to the fourth edition (2008) of Marvin Jay Greenberg's book, which is reference [6].</p> <p>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 <em>The Foundations of Geometry and the Non-Euclidean Plane</em> by George E. Martin, which has a nice little section at the very end. There is also, now, <em>Geometry: Euclid and Beyond</em> by Robin Hartshorne. </p> <p>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.</p> http://mathoverflow.net/questions/43554/is-there-a-compass-and-straight-edge-construction-of-parallel-lines-in-hyperbolic/46622#46622 Answer by Beata Randrianantoanina for Is there a compass and straight edge construction of parallel lines in hyperbolic geometry? Beata Randrianantoanina 2010-11-19T12:57:44Z 2010-11-19T12:57:44Z <p>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.</p>