3D plane with more than 3 points

Hi all,

I am fairly new to matlab-so English please! I am trying to create a 3D plane with approximately 30 x,y,z co-ordinates. Can anyone tell me how I would go about doing this. I can do it with 3 points fine, but anything more and i'm not sure. Appreciate any help. Katie

Answer 1

[Sean de Wolski]

Perhaps ndgrid or meshgrid will help get you started:

[xx,yy,zz] = meshgrid(1:3,4:6,7:9); % grid of x,y,z
V = sin(xx)+cos(yy).^2.*zz;  % some function of x,y,z
isosurface(xx,yy,zz,V,2) % show it

You'll have to provide us with more detail about what you want if you would like more specific info.

[Katie]

Thanks for answering, OK so this is a selection of the points that I have:

x  y  z
690089  7542396  1308
693029  7543757  1508
694311  7544425  1508
695194  7543782  1508
694752  7541297  1508
694319  7536947  1408
694545  7533813  1308
687581  7530576  1308
685263  7528122  1308

Is this what you need?

[Sean de Wolski]

What do you want to do with them?

[Katie]

I need to create a 3D plane that passes through the points.

[Sean de Wolski]

Do you have the Curve Fitting Toolox?

If so, run:

>>cftool

This will allow you to interactively fit all sorts of planes through the points.

[Katie]

OK thanks, I'll give it a go! Probably have some more questions down the line.

[Katie]

The curve fitting tool says that my 'data sizes are incompatible'? Not quite sure why. Part of my data is in seans answer at the top. Thanks

Answer 2

[Matt J]

The code below will generate a plane fit whose equation is dot(N,r)=d where N is the plane's normal.

 P=[x(:),y(:),z(:)];
 c=mean(P);

 P=bsxfun(@minus,P,c);

 [U,S,V]=svd(P);

N=V(:,end);
d=dot(N,c);

[Katie]

Hi Matt, I'm sorry but I'm not very matlab literate. I'm guessing that for the first row I would insert all my values of x,y and z? When [U,S,V]=svd(p) do I need to change anything here. I do not really understand what this is saying?

[Matt J]

I'm sorry but I'm not very matlab literate. I'm guessing that for the first row I would insert all my values of x,y and z?

You told Sean that your x-coordinate data was in a vector called x and similarly for y and z. I assumed that to be the case here as well.

The rest of the code should run as posted.

[Katie]

Hi Matt, when I enter my values it says that 'index exceeds matrix dimensions'. Any ideas?

[Katie]

Scrap that, so I have done the code above, but nothing happens at the end? How do I plot the plane?

[Matt J]

You can plot using the PLOT3 command, the same as in Joseph Cheng's code. However, I don't recommend that you generate the fit as he has done, for reasons I give in the comment to his post.

Answer 3

[Joseph Cheng]

you can always try the Ordinary least squares method to get the coefficients for the plane by the equation Ax=b; A and b are defined below and x will contain the coefficients for you plane.

With your n data points (x[i], y[i], z[i]), the 3x3 symmetric matrix A can be computed by

sum_i x[i]*x[i],    sum_i x[i]*y[i],    sum_i x[i]
sum_i x[i]*y[i],    sum_i y[i]*y[i],    sum_i y[i]
sum_i x[i],         sum_i y[i],         n

Then compute the 3 element vector b:

{sum_i x[i]*z[i], sum_i y[i]*z[i], sum_i z[i]}

Then solve Ax = b for the given A and b by x = A\b;

reading that it's a bit confusing so i've included a quick implementation to show the above math.

x=[1:10];
y=[1:10];
[X Y]=meshgrid(x,y); 
Z = [2*X+3*Y+5+rand(size(X))]; %create a dummy set of x y and z points
%calculate the matrix A
A = [sum(X(:).^2) sum(X(:).*Y(:)) sum(X(:)); sum(X(:).*Y(:)) sum(Y(:).^2) sum(Y(:)); sum(X(:)) sum(Y(:)) length(X(:))]

%calculate b
  b=[sum(X(:).*Z(:)) sum(Y(:).*Z(:)) sum(Z(:))]

%solve for the coefficients
  x = A\b';  %x=[a b c]'.  the equation of the plane will be z = a*x+b*y+c.  
%the coefficients should match closely to the coefficients I used for Z above.
  planefit = [x(1)*X+x(2)*Y+x(3)]; %
  figure,plot3(X(:),Y(:),Z(:),'r.','markersize',20); hold on, surf(X,Y,planefit);

[Matt J]

It's not really a good idea to use Ordinary Least Squares for plane fitting. Your x,y,z data, and hence your A matrix, will have errors in them, too, making the estimator biased. Total Least Squares is better.