adding polar coordinates?

Question

How would you add polar coordinates without converting to cartesian and back.

For simplification, I will use the example of a robot. Lets say I drive 50m with an azimuth of 3rad. I then take a sensor reading at 2rad from the robot and it returns a distance of 10m from the robot to an object.

How do I calculate the objects position from the starting point of the robot without first converting into cartesian, adding the (x,y) pair, and then converting back into polar?

The 2rad is relative to the robots current position and azimuth, not the absolute azimuth from the starting point.

Answer 1

You can apply the Law of Cosines, but you need to clear up the ambiguities first. Is the 2 rad. with respect to the original co-ordinate system (absolute), or is it with respect to the traveled path, and if so, what sense?