Elliptical rotation matrix - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-18T23:33:25Z http://mathoverflow.net/feeds/question/4536 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4536/elliptical-rotation-matrix Elliptical rotation matrix bobobobo 2009-11-07T16:37:18Z 2009-11-07T16:55:12Z <p>We can rotate a point 'circularly' about an arbitrary axis:</p> <p><a href="http://bobobobo.wordpress.com/files/2009/08/rot-arbitrary-axis.png" rel="nofollow">the equation is here, but this site doesn't trust me enough yet to post an image.,</a></p> <p>But as we walk theta 0 -&gt; 2PI this takes the point around a "unit circle" around the axis you're rotating about</p> <p>How can we make it so as theta 0 -&gt; 2PI the results are about an <b>ellipse</b> of width a, height b?</p> <p>I do not want to apply transformation matrices to the points after rotating them about the axis - what I'm looking for is an "elliptical" rotation matrix, if anyone knows of such a thing!</p> http://mathoverflow.net/questions/4536/elliptical-rotation-matrix/4537#4537 Answer by Reid Barton for Elliptical rotation matrix Reid Barton 2009-11-07T16:52:51Z 2009-11-07T16:52:51Z <p>Sure, you can <a href="http://en.wikipedia.org/wiki/Inner_automorphism" rel="nofollow">conjugate</a> the rotation matrix by a matrix which carries the unit circle to the ellipse in question, e.g., the diagonal 2x2 matrix with entries a and b.</p> http://mathoverflow.net/questions/4536/elliptical-rotation-matrix/4538#4538 Answer by Harald Hanche-Olsen for Elliptical rotation matrix Harald Hanche-Olsen 2009-11-07T16:55:12Z 2009-11-07T16:55:12Z <p>Sure. In 2 dimensions:</p> <p>$$\begin{pmatrix}\cos\theta&amp;k\sin\theta\\ -k^{-1}\sin\theta&amp;\cos\theta\end{pmatrix}$$</p> <p>The idea: Scale the $x$ axis by $k$, rotate, then scale back. Now pick $k$ appropriately (left as an exercise).</p>