# Find the error in an orientation angle for noisy points in $R^2$

Assume I have two points in $R^2$:

$P_{t=0} = (x_1, y_1)$ and

$P_{t=1} = (x_2, y_2)$

at $t=1$ I want to calculate the heading angle/orientation of the particle. Both points have measurement noise of N, and the noise is constant.

To calculate the orientation:

$\theta = arctan2((x_{1} - x_{2}), (y_{1} - y_{2}))$

However, I need some uncertainty around $\theta$, that accounts for the noisy x and y values.

I have done the following to try and calculate an error on theta.

$\delta( \ atan2(\Delta x, \Delta y) \ ) = \sqrt{\Sigma_{i} \ \ \big(\delta(a_i) . \frac{d(atan2(\Delta x, \Delta y))}{da_i}}\big)^2$

Where: $a_i \ \epsilon \ [\Delta x, \Delta y]$

$\Delta x = x_1 - x_2 \ \ \$ $\Delta y = y_1 - y_2$

$\frac{d(atan2(\Delta x, \Delta y))}{da_i}$ is the partial derivative of atan2.

$\delta(a_i)$ is effectively the noise on $\Delta x \ or \ \Delta y$

i.e. $\delta(\Delta x)$ or $\delta(\Delta y)$ depending on $i$.

$\delta(\Delta x) = \sqrt{ N^2 + N^2 }$

I DO NOT propagate error when calculating the partial derivative of arctan2. Should I be doing this?

I tried to test this error on $\theta$, by computing a Gaussian of theta and the error. In about ten random samples of this distribution, I get perhaps one value that is indeed close to theta. But it seems a bit poor - not good enough of an error for theta.