I've been trying to study the error on the center point between two skew lines, i.e. the point at mid-distance between the two lines on their common perpendicular.
The two lines are each defined by two known points, a, b and a', b'.
I first found the equation of the center point by solving the system made of the scalar products between the common perpendicular direction vector and each line's direction vector, which are both equal to zero. So far everything went fine.
Then I tried to derive the error on that center point, by applying the error propagation formula to all equations. I was pretty careful not to forget a term or an absolute value. At this point I think I've checked my math 10 times thoroughly.
The results were pretty OK, until I tried to study the sensitivity of this error w.r.t. the angle between the two lines. For example, if I take two line perpendicular to each other, and with no intersection, defined by these points:
a = [10 5 10]; b = [-10 5 -10]; a' = [-10 -5 10]; b' = [10 -5 -10];
Each line is at +/-45° in the xz-plane. Lets say in that situation the error is [0.7 0.2 0.7]. If I change this angle to 35° (flatten the lines) I get an error of [1.3 0.2 0.99]. If instead I change the angle to 55°, I expect to get an error of [0.99 0.2 1.3], since the problem is symmetric.
However, instead of [0.99 0.2 1.3], I get an error of [0.96 0.5 1.28] for an angle of 55°. The trend is correct but the values are slightly off.
At this point I've run out of ideas for where this difference comes from. I've implemented my math in excel and in Matlab with similar results.
If anyone has an idea of what could be the source of error, or if you feel like giving it a try yourself, that would be greatly appreciated.
Cheers !