Finding the midpoint of intersection for multiple planes

Question

I have some planes in three-dimensional space that theoretically should intersect on the same line.

Each plane is calculated by taking 3 measured points. However, there is some error in measuring these points.

Thus, in practice, not all planes intersect on the same line.

How can I get the "middle" line of intersection for all of these planes?

I could just cross each plane with each other by a plane and average the lines. However, when two planes have almost the same orientation, a small error in their measurements will lead to a huge error in their calculated intersection. Therefore, when calculating the average intersection in this way, I would increase the measurement error.

Is there a fairer way to calculate a common intersection line?

Answer 1

I think that you need not averaging, but an algorithm that detects outliers (i.e. bad intersection lines) and discards them.

One standard algorithm that does this is random consensus ( RANSAC ). But in order to use it, you would need to determine some distance between the lines, which should be possible to make a meaningful way.

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