plot 3D line, matlab

Question

My question is pretty standard but can't find a solution of that.

I have points=[x,y,z] and want to plot best fit line.

I am using function given below (and Thanx Smith)

% LS3DLINE.M   Least-squares line in 3 dimensions.
%
% Version 1.0    
% Last amended   I M Smith 27 May 2002. 
% Created        I M Smith 08 Mar 2002
% ---------------------------------------------------------------------
% Input    
% X        Array [x y z] where x = vector of x-coordinates, 
%          y = vector of y-coordinates and z = vector of 
%          z-coordinates. 
%          Dimension: m x 3. 
% 
% Output   
% x0       Centroid of the data = point on the best-fit line.
%          Dimension: 3 x 1. 
% 
% a        Direction cosines of the best-fit line. 
%          Dimension: 3 x 1.
% 
% <Optional... 
% d        Residuals. 
%          Dimension: m x 1. 
% 
% normd    Norm of residual errors. 
%          Dimension: 1 x 1. 
% ...>
%
% [x0, a <, d, normd >] = ls3dline(X)

I have a. So equation may be

points*a+dist=0

where dist is min. distance from origon.

Now my question is how to plot best filt line in 3D.

Answer 1

It helps to actually read the content of the function, which uses Singular Value Decomposition.

% calculate centroid
  x0 = mean(X)';

% form matrix A of translated points
  A = [(X(:, 1) - x0(1)) (X(:, 2) - x0(2)) (X(:, 3) - x0(3))];

% calculate the SVD of A
  [U, S, V] = svd(A, 0);

% find the largest singular value in S and extract from V the
% corresponding right singular vector
  [s, i] = max(diag(S));
  a = V(:, i);

The best orthogonal fitting line is

P = x0 + a.*t

as the parameter t varies. This is the direction of maximum variation which means that variation in the orthogonal direction is minimum. The sum of the squares of the points' orthogonal distances to this line is minimized.

This is distinct from linear regression which minimizes the y variation from the line of regression. That regression assumes that all errors are in the y coordinates, whereas orthogonal fitting assumes the errors in both the x and y coordinates are of equal expected magnitudes.

[Credit: Roger Stafford , http://www.mathworks.com/matlabcentral/newsreader/view_thread/294030]

Then you only need to create some t and plot it:

for t=0:100,
P(t,:) = x0 + a.*t;
end
scatter3(P(:,1),P(:,2),P(:,3));

You may want to use plot3() instead, in which case you need only a pair of points. Since a line is infinite by definition, it is up to you to determine where it should begin and end (depends on application).