\chapter{projection equations}

The equation for a point object located at ($x_0$, $y_0$, $z_0$) in a circular scan trajectory is given by:
\begin{equation}\label{eq:circularorbit}
\begin{split}
x_{\alpha}& = x_0 \cos \alpha \pm y_0 \sin \alpha, \\
y_{\alpha}& = \mp x_0 \sin \alpha + y_0 \cos \alpha, \\
z& \equiv z_0.
\end{split}
\end{equation}
On a perfectly well aligned detector the projection orbit of the point, $(u_{\alpha}^{(id)}, v_{\alpha}^{(id)})$, in the (x,y,z) coordinate is then given by its projections onto the detector plane, 
\begin{equation}\label{eq:idealUVpoints}
u_{\alpha}^{(id)} = \frac{R x_{\alpha}}{y_{\alpha} + R_F}, \hspace{0.4cm} v_{\alpha}^{(id)} = \frac{Rz_0}{y_{\alpha}+R_F}, 
\end{equation}
where $R$ and $R_F$ are the distances from the focus to the detector and the axis of rotation, respectively.  For a misaligned detector the point (u,v) with (x', y', z') coordinates, the point locations on the detector is given by:

\begin{equation}\label{eq:circularorbit}
\begin{split}
x_{\alpha}& = x_0 \cos \alpha \pm y_0 \sin \alpha, \\
y_{\alpha}& = \mp x_0 \sin \alpha + y_0 \cos \alpha, \\
z& \equiv z_0.
\end{split}
\end{equation}
On a perfectly well aligned detector the projection orbit of the point, $(u_{\alpha}^{(id)}, v_{\alpha}^{(id)})$, in the (x,y,z) coordinate is then given by its projections onto the detector plane, 
\begin{equation}\label{eq:idealUVpoints}
u_{\alpha}^{(id)} = \frac{R x_{\alpha}}{y_{\alpha} + R_F}, \hspace{0.4cm} v_{\alpha}^{(id)} = \frac{Rz_0}{y_{\alpha}+R_F}, 
\end{equation}
where $R$ and $R_F$ are the distances from the focus to the detector and the axis of rotation, respectively.  For a misaligned detector the point (u,v) with (x', y', z') coordinates, the point locations on the detector is given by:
\begin{equation}\label{eq:nonidealUVpoints}
\begin{split}
\begin{pmatrix}
u \\ v 
\end{pmatrix} = & \frac{1}{\det \mathbf{Q}} 
\begin{pmatrix}
o_{33}-o_{23}v^{(id)}/R & -(o_{13} - o_{23} u^{(id)}/R) \\
-(o_{31} - o_{21} v^{(id)}/R) & o_{11}-o_{21} u^{(id)}/R \\
\end{pmatrix} \\
\times & 
\begin{pmatrix}
u^{(id)'} - d_x \\
v^{(id)'} - d_z 
\end{pmatrix},
\end{split}
\end{equation}
where the primes denote a simple rescaling of the ideal orbit by
\begin{equation}
\begin{pmatrix}
u^{(id)'} \\ v^{(id)'}
\end{pmatrix}
= \frac{R_y'}{R}
\begin{pmatrix}
u^{(id)} \\ v^{(id)}
\end{pmatrix}.
\end{equation}
where
\begin{equation}
\begin{split}
\det \mathbf{Q} =& (o_{11} - o_{21} u^{(id)}/R)(o_{33}-o_{23}v^{(id)}/R) \\
& -(o_{13}-o_{23}u^{(id)}/R)(o_{31}-o_{21} v^{(id)}/R).
\end{split}
\end{equation}
and $\mathbf{O}$ is the rotation matrix of the misaligned detector shown by 
\begin{equation}
\mathbf{O} = 
\begin{pmatrix}
\cos \eta \cos \varphi - \sin \eta \sin \theta \sin \varphi & -\cos \theta \sin \varphi & -\cos \varphi \sin \eta - \cos \eta \sin \theta \sin \varphi \\
\cos \varphi \sin \eta \sin \theta + \cos \eta \sin \varphi & \cos \theta \cos \varphi & \cos \eta \cos \varphi \sin \theta - \sin \eta \sin \varphi \\
\cos \theta \sin \eta & -\sin \theta & \cos \eta \cos \theta
\end{pmatrix}
\end{equation}