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I am confused on a very simple thing, so I need your clarifications. Here is my doubt: I want to find the intersection point of three straight lines. Alternatively, I can find it by using three planes. (Assuming I derived the planes from some point data and contain some outliers)

So for the first case (i.e. for 3 lines case) I think I can first find the intersection point of 2 lines and then take another 2 and so on. then, may be I will get 3 points. Hence, I need some sort of adjustment (e.g. least square) to get the correct point. If I take the 3 planes intersection case, I think I can first select 2 planes and then I can find the intersection line and then get the third plane. Finally, line-plane intersection would give me the relevant point. Maybe the methods that I mentioned are not correct.

So, now I want to know what is the most precise method to get intersection point from

  1. three lines
  2. three planes

and what is most precise or both are equal? many thanks

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up vote 1 down vote accepted

If you are in $\mathbb{R}^3$, two lines will generally not intersect. You can find the point that minimizes the sum of the (squares of the) distances to the three lines. Information of how to calculate the distance to the line in three dimensions is at Mathworld, then use a function minimizer. In general three planes will intersect in a point. You can solve the simultaneous equations of the planes to find it. If there are errors in your planes, there will be an error in your point.

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In general, three planes (in ${\bf R}^3$) will intersect in a point. – Gerry Myerson Jul 28 '11 at 1:59
Or, to recast Gerry's comment in different terms: "most" (for some rigorous definition of "most") $3\times 3$ matrices are invertible. – J. M. Jul 28 '11 at 2:29
@Gerry Myerson: you are right. Failure of visualization. Fixed – Ross Millikan Jul 28 '11 at 3:57
@ Ross Millikan>> thanks for the comment. (Me, poor in geometry). So, need more help. Since i don’t have much space here, i will split my post. Please look all those. – niro Jul 28 '11 at 9:27
@ Ross: Suppose i have 3 (or more) lines. So, I will get ONE or MORE intersection points based on the number of lines. But I need ONE point at the end. So.. I am following your statement:- "You can find the point that minimizes the sum of the (squares of the) distances to the three lines." but, this mathworld gives distance between 2 lines. – niro Jul 28 '11 at 9:31

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