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May 16th, 2011, 06:22 AM   #1
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Fit a plane to 3D cartesian points query?

Hi Everyone

I'm a computational Biologist whose trying to fit a collection of 3D cartesian co-ordinates to a plane, using just the cartesian co-ordinates what is the best method to fit the plane to the data points?

Many Thanks

Best Wishes

Dan
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May 16th, 2011, 01:58 PM   #2
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Re: Fit a plane to 3D cartesian points query?

Use least squares. Let the plane be ax + by + cz + d = 0. For each point get the expression for the distance from the point to the plane. Let f(a,b,c,d) = sum of the squares of the distances. Find a,b,c,d which minimizes f.
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May 17th, 2011, 12:39 PM   #3
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Re: Fit a plane to 3D cartesian points query?

Thanks so much!

Sorry to be a pain but would you mind giving an example with working, my maths is seriously rusty:

e.g: with say

point 1: x = 2.73 y = -3.28 z = 19.45
point 2: x = 6.33 y = 43.28 z = 3.45
point 3: x = 2.83 y = 11.42 z = 92.67
point 4: x = 19.92 y = 95.88 z = 8.92

many thanks

Dan
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May 17th, 2011, 12:49 PM   #4
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Re: Fit a plane to 3D cartesian points query?

Frazierangus,

Building on the method suggested by [color=#00BF00]mathman[/color], I will just give you the formula for the distance D between a point P(x?,y?,z?) and the plane ax + by + cz + d = 0:



I hope this helps.

-Ormkärr-
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May 17th, 2011, 01:24 PM   #5
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Re: Fit a plane to 3D cartesian points query?

For least squares, use D^2 rather than D, thus avoiding | | in the numerator.
Furthermore there is one free variable in the definition of the plane, so the arithmetic is greatly simplified by assuming a^2 + b^2 + c^2 = 1.

What I mean by free variable is that multiplying the four parameters (a,b,c,d) by a constant doesn't change the plane.
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