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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

1 Answers 1

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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.2011-07-28
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    Or, to recast Gerry's comment in different terms: "most" (for some rigorous definition of "most") $3\times 3$ matrices are invertible.2011-07-28
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    @Gerry Myerson: you are right. Failure of visualization. Fixed2011-07-28
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    @ 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.2011-07-28
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    @ 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.2011-07-28
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    @ Ross: but for me, i have many lines. so how can i get final point. is it by using least square or by modifying the sum of square distances method for many line-point pairs? Need some hints, equations for this.2011-07-28
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    @ Ross: Also, not clear "function minimizer". What is this? is this the part d(d^2)/dt=0 mentioned in the same mathworld page. (Actually i derived my lines by intersecting 2 planes at a time. so i guess, i don’t want to follow - solving simultaneous plane equations method to get final intersection point.). if you can explain all the steps that i should follow, it is a great help.Many thanks2011-07-28
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    If you have lots of lines, the procedure is the same. You want to find the point that is closest to the lot of them, and sum of distances is a good metric. The function minimizers are numeric routines. My favorite source (any numerical analysis book will have it) is chapter 10 of http://www.nr.com/ The obsolete versions are free on line.2011-07-28
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    @ Ross>> many thanks for the clarifications. if you dont mind i wan your help again:..if you know, please let me know some other web sites which i can refer (as it cannot refer Numerical recipies website freely). may be you could simply tell how i can implement this for many line case, i am really poor in geometry. really i dont have any idea about what the " numeric routines or function minimizers". please give some explanations.. thank you.2011-07-29
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    If you go to nr.com, click on Obsolete Versions on the left, then under Empanel interface, Numerical Recipes in C (1992), you can see the whole book for free. I bought it 30 years ago and find the explanations excellent.2011-07-29
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    thanks.i got the chapter 10. there are many ways..difficult to think what is best for this case. dear ross, please point out the main steps to get final intersection point from the lines as i am not sure how it would be.2011-07-31
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    Any of the ones in 10.4 to 10.7 should work, as your problem is not numerically difficult. The simplex method, 10.4, is probably simplest, as it does not require you to calculate derivatives.2011-07-31