My Math Forum Find a fitting plane to given 3d points

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 December 1st, 2010, 01:20 PM #1 Newbie   Joined: Dec 2010 Posts: 2 Thanks: 0 Find a fitting plane to given 3d points Hi All, This is my first post so pardon me if it's in the wrong place. I am developing a solution where I need to find a plane that is fitted to given points in 3d space. I tried looking at orthogonal distance regression, but I found it a bit difficult to understand. Here is one link (http://mathforum.org/library/drmath/view/63765.html) that I found quite helpful, however I am stuck when the matrix form begins. Can someone help me with either understanding that link or give some alternative solutions? I appreciate any help. Thanks. -Maulik
 August 28th, 2011, 10:52 PM #2 Senior Member   Joined: Aug 2011 Posts: 334 Thanks: 8 Re: Find a fitting plane to given 3d points Hello ! This is a link to a paper dealing with 3-D. Linear Regression and 3-D. Plane Regression : http://www.scribd.com/people/documents/ ... jjacquelin Then select "Regressions et trajectoires en 3D". A non-recursive algorithm for orthogonal mean squares fitting is ginen page 21 (Updated edition July 11, 2011)
 August 29th, 2011, 04:27 AM #3 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Find a fitting plane to given 3d points Any (non-vertical) plane can be written as z= Ax+ By+ C. Given points $(x_1, y_1, z_1)$,$(x_2, y_2, z_2$, ..., $(x_n, y_n, z_n)$ we can think of this as solving the system of equations $Ax_1+ By_1+ C= z_1$, $Ax_2+ By_2+ C= z_2$, ... , $Ax_n+ By_n+ C= z_n$. And we can cast that into a matrix problem: $\begin{bmatrix}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ . & . & . \\ . & . & . \\ . & . & . \\ x_n & y_n & 1\end{bmatrix}\begin{bmatrix}A \\ B \\ C\end{bmatrix}$$= \begin{bmatrix}y_1 \\ y_2 \\ ... \\ y_n\end{bmatrix}$ Well, of course, if n> 3, this is problematical. We can think of this as "Ax= y" where the matrix, A, maps all of $R^3$ into a 3 dimensional subspace of $R^n$. If y does not happen to be in that subspace there is no such x. But we can ask for the x such that Ax is closest to y. Of course, in that case, the difference, Ax- y, must be perpendicular to the subspace. That is, = 0 for any u in $R^3$. Now, a little theory. If A is a linear transformation from one inner product space, U, to another, V, then its adjoint, A*, is the linear transformation from V back to U such that = for all u in U, all v in V. Notice that the inner product on the left is in U, the inner product on the right is in V. It is easy to show that if A is from $R^m$ to $R^n$, written as a matrix, its adjoint is just its transpose. Here, we can go from = 0 to = 0. The important difference is that now the innerproduct is in $R^3$ and u could be any vector in that space. The only vector that has inner product 0 with all vectors (including itself) is 0. We must have A*(Ax- y)= A*Ax- A*y= 0. Note that A is from $R^3$ to $R^n$, so a "3 by n" matrix, while A* is from $R^n$ to $R^3$, written as an "n by 3" matrix so that A*A is a 3 by 3 matrix. There are circumstances in which even A*A does not have an inverse but typically it does. Assuming that, we have, from A*Ax= A*y, x= (A*A)^{-1}A*y. Here, as before, $\begin{bmatrix}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ . & . & . \\ . & . & . \\ . & . & . \\ x_n & y_n & 1\end{bmatrix}$ so that $A*= \begin{bmatrix} x_1 &x_2=&...=&x_n \\ y_1=&y_2=&...=&y_n \\ 1=&1=&...=&1\end{bmatrix}=$ and $A*A= \begin{bmatrix}\sum x_i^2 &\sum x_iy_i=&\sum x_i \\ \sum x_iy_i=&\sum y_i^2=&\sum y_i \\ \sum x_i=&\sum y_i=&n\end{bmatrix}=$. That last is a symmetric 3 by 3 matrix and fairly easily invertible so it is not at all difficult to fine $x= (A*A)^{-1}A*y$ where x, of course, is the vector $\begin{bmatrix}A \\ B\\ C\end{bmatrix}$ of coefficients of the desired linear function.
 August 29th, 2011, 05:39 AM #4 Senior Member   Joined: Aug 2011 Posts: 334 Thanks: 8 Re: Find a fitting plane to given 3d points Hi , HallsofIvy , as far as I can understand, maulik13 asked for orthogonal distance regression. The method that you propose is not an orthogonal one.
 September 20th, 2011, 02:03 PM #5 Newbie   Joined: Dec 2010 Posts: 2 Thanks: 0 Re: Find a fitting plane to given 3d points Hi Guys! Thank you for you replies. I will have to study your replies and I will post in a bit more detail. And yes I was looking for orthogonal regression analysis.

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