Calculating the intersection volume of 2 3D volumes defined by point and accuracy

Question

I'm surprised this hasn't been asked before - it may be that my ability to search the site is hampered by being on the app!

I have $2$ 3D points $(X_1, Y_1, Z_1)$ and $(X_2,Y_2,Z_2)$ each with associated accuracy in those dimensions $(dX_1,dY_1,dZ_1)$ and $(dX_2,dY_2,dZ_2)$. I'm treating these accuracies as a volume in which the points exist. I'd like to try and calculate:

1. If the volumes intersect,
2. The volume of the intersection if they do intersect.

I know how to calculate is two lines intersect but I have no idea how to transfer that over to 3D volumes!

Answer 1

1. The condition for X: $\left | X_{1}-X_{2} \right |< dX_{1}+dX_{2}$. Now extend it to y and z. All of them need to be fulfilled simultaneously to get a intersection.

2. The intersection is rectangular prism on its own. You need to measure the overlap for each dimension separately. This is accomplished by first by comparing the values of $X_{1}$ to $X_{2}$ and properly use $dX_{i}$ to calculate the overlap for X dimension. Than do the same for Y and Z. I assume you know how to do the overlap computation.