PDF of distance between measured coordinates with normally distr. errors

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hi,
the problem i have is this:

consider a line segment L of length d in R^3. the projection of the endpoints of L on a fixed plane can be determined with an error that is normally distributed, and the errors for the two endpoints needs not to be equal. Assuming that L is randomly oriented in R^3, what is the distribution of the projected distance?

I have been thinking about this for quite a while, reading up on chi(-squared), rice, reighley distributions. the problem is that the coordinates are not independent, so that dX,dY in sqrt(dX^2+dY^2) are not independent, making these distributions unsuitable. The PDF of the projected distance neglecting contribution of errors is

r/d*1/sqrt(d^2-r^2)

where r is the projected distance on the plane.

Thanks for all the help
 

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