# Standard Deviation from one axis to another axis

I have radar mounted on a car. For each detection, the radar returns these variables.

• relative distance between the object/host vehicle in forward direction
• standard deviation of the relative forward distance
• relative distance between the object/host vehicle in left/right direction
• standard deviation of the relative left/right distance

I'm trying to do coordinate transform of the above data, so that I get relative distance/standard deviation in global coordinates (North, South, East, West) Distance is easy since it only requires to rotate the axis by the amount of angle between the vehicle body-fixed axis and the global axis.

How about standard deviation? How do I transform the standard deviation from the vehicle body-fixed axis to global axis?

Standard deviation is used to represent probability distribution - in your case if you have $\sigma = 1m$ for forward distance it means that there is ~68% chance of true forward distance to the obstacle be less than 1m away from your measurement. You have two variables, so you should project your probability distribution on a 2D surface - you will get something like that:
In general you can find your covariance matrix for global coordinate system with: $\Sigma = R S S R^{-1}$ where $R$ is rotation matrix from local to global coordinates and $S$ is diagonal matrix with your variances as diagonal elements.