# Intersection of 3 rotated non-orthogonal planes

Here's my problem. Say I have three orthogonal planes (one with a normal vector along the $x$-direction, one along the $y$-direction, and one along the $z$-direction). Initially they intersect at $(x_0,y_0,z_0) = (0,0,0)$. First I rotate these planes by some known (non-orthogonal) rotation matrix, to come up with 3 new planes (all of this is easy, i'm getting to my tricky part). Lets assume that none of the planes are parallel, so we get a single intersection point. I can then shift these planes along their normal vector (which can easily be done by adding some fraction of the normal vector to the plane's point). I can now easily find the new intersection point by using the equation given here: http://mathworld.wolfram.com/Plane-PlaneIntersection.html .

Here's my question, instead of finding the intersection point, I want to instead find the values $a_1, a_2, a_3$ (the coefficients of shifting that translate my three rotated matrices) so that my end point (after i have rotated the matrices), is a given point $(\bar{x},\bar{y},\bar{z})$. Basically it's an inverse problem. Instead of having 3 planes with known shifts and translates and finding the intersection point, i know the rotation, and the intersection point, but i don't know the shift values. Any help would be greatly appreciated to get me going on the right path. Thanks very much!

• non-orthogonal rotation matrix - I'm not positive what this means. What kind of rotation isn't orthogonal? Maybe you are saying rotations can occur about any point (not just the origin?) – rschwieb Sep 19 '13 at 19:23
• ah ok, maybe I didn't word this properly. What I mean is each of the 3 planes are being rotated by their own rotation matrix (which is, itself orthogonal, but all 3 are different). You're right to say non-orthogonal didn't make sense, there's a reason why i wrote that but ignore it cause it's misleading and wrong in my context. – Karl Landheer Sep 19 '13 at 20:26
• But the three rotations all take place leaving the origin fixed, right? – rschwieb Sep 19 '13 at 23:23