Combining general 1D normal distributions into a 2D distribution

My question is a generalization of the question asked here

There is a point in 2-D space. I can measure the range of this point from two other locations. I get this measurement as a mean (range) and standard deviation (error) of a normal distribution. What I would like is to transform these two 1-D normal curve in a 2-D normal distribution that will give me the mean location and error (as a 2x2 co-variance matrix)

In the referred question, the two directions are constrained to be normal to each other. My question is that if the directions are any two general directions (given by 2D vectors) and associated with each is the uncertainty in the form of a standard deviation value, can we still use the solution given in that question? Probably not, as any two general vectors will not be eigenvectors. What will be the solution then?

Secondly, if the mean of both measurements is NOT zero (as taken in the solution to referred question), how will we go about the solution.

Thirdly, can anyone point me to additional reading on the solution provided by Michael Hardy in response to the referred question.

Thanks in advance

• would you mind to make the question a little more self-contained? it's ok to reference another question, but at least a basic statement... – leonbloy Jun 4 '12 at 20:40