Let's assume a camera in space and an image point in this camera:
$t \in \mathbb{R}^3$ is the position of the camera in space.
$R \in \mathbb{R}^{3 \times 3}$ is the orientation of the camera in space.
$K \in \mathbb{R}^{3 \times 3}$ is the camera intrinsic matrix.
$M = (R^T|-R^Tt) \mathbb{R}^{3 \times 4}$ is the camera extrinsic matrix.
The projection in homogenous coordinates between the camera and real world is:
$\begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = K M\begin{bmatrix} X \\ Y \\ Z \\ 1 \end{bmatrix}$
Consider a plane $E = \{x \in \mathbb{R}^3| x^T n=0 \}$ with normal vector $n \in \mathbb{R}^3$.
Let's consider the case where the ray $L = \{\lambda\begin{bmatrix} X \\ Y \\ Z \end{bmatrix} | \lambda \in \mathbb{R}^3 \}$ has a unique intersection with $E$, the surface point $S \in E \cap V$.
Let's say $R = R_1(\varphi_{roll}) R_2(\varphi_{pitch}) R_3(\varphi_{yaw})$.
Now we assume angular errors $\phi_{roll}, \phi_{pitch}, \phi_{yaw} \sim N(0, \sigma_{roll}), N(0, \sigma_{pitch}), N(0, \sigma_{yaw})$.
And spacial errors $T_x, T_y, T_z \sim N(0, \sigma_x), N(0, \sigma_y), N(0, \sigma_z)$.
Without loss of generality we may assume $E = \mathbb{R}^2 \times \{0\}$ and we can identify $S \in \mathbb{R}^3$ as $S' \in \mathbb{R}^2 $.
I am interested in an approximate computation of the covariance of $\Sigma_{S'}$, which is the uncertainty that results in the projection of the image point to the surface point $S'$, which also becomes a random variable if the position and angles of the observing camera are random. We can assume that all angular errors are small and approximate Taylor series can be used for our purposes.