My Math Forum Fit a plane to 3D cartesian points query?

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 May 16th, 2011, 06:22 AM #1 Newbie   Joined: May 2011 Posts: 3 Thanks: 0 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
 May 16th, 2011, 01:58 PM #2 Global Moderator   Joined: May 2007 Posts: 6,807 Thanks: 717 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.
 May 17th, 2011, 12:39 PM #3 Newbie   Joined: May 2011 Posts: 3 Thanks: 0 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
 May 17th, 2011, 12:49 PM #4 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 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: $D\=\ \displaystyle\frac{\left|ax_0+by_0+cz_0+d\right|}{ \sqrt{a^2+b^2+c^2}}.$ I hope this helps. -Ormkärr-
 May 17th, 2011, 01:24 PM #5 Global Moderator   Joined: May 2007 Posts: 6,807 Thanks: 717 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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