Mapping from $\text{PSL}(2,\mathbb{R})$ to transformations of the hyperboloid - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:54:55Z http://mathoverflow.net/feeds/question/105490 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105490/mapping-from-textpsl2-mathbbr-to-transformations-of-the-hyperboloid Mapping from $\text{PSL}(2,\mathbb{R})$ to transformations of the hyperboloid Aaron Golden 2012-08-26T01:03:09Z 2012-08-26T01:16:05Z <p>Let $F$ map points in $\mathbb{R}^3$ to points on the unit disk, $\Delta$, in the $xz$-plane (identified with $\mathbb{C}$) by projecting through $\Delta$ along lines that intersect at $(0,-1,0)$. Let $H$ be the forward sheet of the hyperboloid as in the <a href="http://en.wikipedia.org/wiki/Hyperboloid_model" rel="nofollow">hyperboloid model</a>.</p> <blockquote> <ol> <li><p>Given an element $p:z\to(az+b)/(cz+d)$ of $\text{PSL}(2,\mathbb{R})$ is there a transformation $g$ of $H$ such that $F \circ g \circ F^{-1} = p$ on $\Delta$?</p></li> <li><p>If the answer to (1) is "yes," then is there a known explicit formula for $g$ in terms of $a$, $b$, $c$, and $d$?</p></li> </ol> </blockquote> <p>To clarify the situation, in the attached figure $g$ is a transformation of $H$ that nearly satisfies the condition from (1) on the green test curve. I found $g$ in the figure by manually tweaking the elements of a 4x4 matrix until the red (projected) and green (correct) curves became visually close in the disk on the bottom right.</p> <p><img src="http://i.imgur.com/wp1W1.png?1" alt="Figure showing transformations between the transformed hyperboloid and the unit disk"></p>