Using theta and Ro you can determine the 2 dimensional position of the center of the cross section using the following and solving for x and y
Sin theta = y/Ro
Cos theta = x/Ro

The same math would determine the 2 dimensional position at the end of Ri on the cross section
Sin phi = y/Ri
Cos phi = x/Ri (this would also be the Z position in the 3d representation)

Finding the 3d x and y positions is a bit more complex and to be completely honest, it's late enough in the day that I don't want to think that hard.

Edit, re-read your post. So theta is given and the x/z/y coordinates where Ri hit the surface are given?