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 rkaminski November 2nd, 2010 12:03 PM

Detector geometry (affine space)

Hi All,

I have some problem with my calculations of the area X-ray detector geometry. Below I present my problem in details.

Assume we have two Cartesian coordinate systems (bases) in the 3D space, $B=(o;\vec{e}_x,\vec{e}_y,\vec{e}_z)$ and $B'=(o';\vec{e}'_x,\vec{e}'_y,\vec{ e}#39;_z)$. They are anchored in points $o$ and $o'$, respectively. Base $B'$ is constructed from $B$ in such a way: (1) basis vectors are rotated by $X$, $Y$ and $Z$ axes (by a small angles, these are detector tilt angles); (2) the coordinate system is moved by distance $d$ along $X$ axis; (3) it is moved along $X$, $Y$ and $Z$ axes by $dx$, $dy$ and $dz$ distances, respectively (these are mechanical offsets which I'd like to consider).

Now, I have a unit vector $\vec{s}$, anchored at point $o$. I would like to calculate the point of intersection of elongation of this vector (i.e. $\vec{p}=p\cdot\vec{s}\$) with the $Y'Z'$ plane (defined by vectors $\vec{e}'_y\$ and $\vec{e}'_z$). I was trying to calculate this but I'm completely stuck...

Any help is greatly acknowledged.

Regards,