most recent 30 from http://mathoverflow.net 2013-06-19T22:14:08Z http://mathoverflow.net/feeds/question/20196 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20196/is-there-any-algorithm-for-determining-3d-position-in-such-case Ole Jak 2010-04-02T22:23:30Z 2010-04-17T01:22:17Z

<p>Suppose I have the following image (i.e. I have the coordinates of all points in 2d so I can regenerate lines and check where they cross each other)</p> <p><img src="http://superior0.narod.ru/2d.jpg" alt=""></p> <p>Now suppose I have another image of what I know to be the same lines:</p> <p><img src="http://superior0.narod.ru/3d.jpg" alt=""></p> <p>How can I determine plane rotation and Z depth on second image (asuming first one's center was in point (0,0,0) with no rotation)?</p>

http://mathoverflow.net/questions/20196/is-there-any-algorithm-for-determining-3d-position-in-such-case/20202#20202 Dror Speiser 2010-04-03T00:13:42Z 2010-04-03T00:13:42Z

<p>What you see in the second image is a projection of the first, after a rotation. So treat it exactly like that. Meaning, you have 4 lines in $\mathbb{R}^3$. You know their equations. Furthermore, you have another 4 lines in $\mathbb{R}^2$ corresponding to the second image.</p> <p>To solve, parameterize the family of possible 4 lines in $\mathbb{R}^3$ that project onto the 4 lines in $\mathbb{R}^2$. Find the matrix that takes the original 4 lines to generic quadruple in the mentioned family. Now write equations to ensure that matrix is actually a rotation. You should have enough information for there to be at most one quadruple that can actually be gotten from the original lines. If not, then your question has multiple answers.</p>