MATLAB: Polinomial approximation of surface

MATLABpolyfitsurface approximation

I have a massive of [x y z] coordinates of points of some surface. I need to get the polynomial approximation of this surface. Standard Matlab function polyfit does not approach like others. I have found polyfitn (+ polyvaln) function via file exchange which applies to the similar task, but it does not satisfy me.

All the functions that I have found alloy to approximate surface with formula z = f(x,y), but I need to get f(x,y,z) = 0. Approximate polynomial will be like this:

A11*x^2 + A22*y^2 + A33*z^2 + A12*x*y + A13*x*z + A23*y*z + A1*x + A2*y + A3*z + A0 = 0;

in case of 2nd order. Can anyone help me?

Best Answer

You can try using matlab's 'svd' function. It will find a set of coefficients A11, A22, A33, ...,A0 for which the sum of the squares of the left sides of your quadratic expression for each of your (x,y,z) points is a minimum subject to the constraint that the sum of the squares of the coefficients be one. There are many ways of defining a "best" fit but this is one definition that has an easy solution.

Let X, Y, Z be column vectors for your given points. That is, the n-th elements of the three vectors are, respectively, the x, y, and z coordinates for the n-th point. Then form the matrix M and use 'svd' to do a singular value decomposition of M:

 M = [X.^2,Y.^2,Z.^2,X.*Y,X.*Z,Y.*Z,X,Y,Z,ones(size(X))];
 [~,S,V] = svd(M,0);
 A = V(:,10);

The vector A which is the rightmost column of V will contain the desired list of coefficients, A11 A22, etc. It is a normalized eigenvector and the sum of its squares will accordingly be one. The smallest singular value that occurs at the lower right side of the main diagonal of the 10-by-10 matrix S, S(10,10), can be considered a measure of the quality of fit.

Related Solutions

MATLAB: How to fit custom polynomial equation

You could download polyfitn from the file exchange.

But if you already know how to use the curve fitting toolbox, then just specify a custom equation of the desired form. Something like this:

modelform = fittype('p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2 + p30*x^3 + p21*x^2*y',...
    'coeff',{'p00','p10','p01','p20','p11','p02','p30','p21'},...
    'indep',{'x','y'},'depend','z');

It will be treated as a nonlinear model internally as a function of the parameters, but since all of the coefficients actually do enter linearly, they will be easy to fit. fit should have no problems.

MATLAB: How to find the equation of the line that intersects a 3D surface and a plane passing through 3 points of interest on the surface

While there is no way of directly getting the intersecting line's equation from the Surface Fitting Tool. Achieving this will require a combination of symbolic analysis (by hand) and regression analysis in MATLAB.

To start with, you will need at least 3 points in order to project a 3-dimensional surface onto 2 dimensions. With only two points, there are an infinite number of curves that can simultaneously intersect the two points along the surface of interest.

First, we create an example dataset to work with. Here, we will try to fit a 2-dimensional second degree polynomial (a surface).

   [x,y] = meshgrid(-2:0.1:2);
   z = x.^2 + 3 + y.^2;
   [xi,yi,zi] = deal(x(:),y(:),z(:))

Open the Surface Fitting Tool using the 'sftool' command and perform the fit using the vectors xi, yi, zi that we just created. This should yield an equation for the surface as follows, where x and y are the variables.

 Linear model Poly22:
        f(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2

We know the 3 points of interest that define the intersecting plane, defined in the form 'z = ax + by + c', where a, b, and c are unknowns that we need to find next. Let the coordinates for these 3 known points be defined as:

 Point1 :  [x1, y1, z1]
 Point2 :  [x2, y2, z2]
 Point3 :  [x3, y3, z3]

To get the equation of the plane, we solve for the coefficients of the plane by regressing using the MRDIVIDE ("\") operator:

W = [x1 y1 1;
           x2 y2 1;
           x3 y3 1];
z = [z1; 
         z2;
         z3];
plane = W\z;
a = plane(1);
b = plane(2);
c = plane(3);

Now, we know the equation of the intersecting plane:

     fplane(x,y) = z = a*x + b*y + c

To find the equation of the line that intersects between the sphere and the plane, we can equate the two expressions, f(x,y) for the surface and fplane(x,y) for the plane that we found, and simplify by hand or simplify by using the Symbolic Math Toolbox.

     fplane(x,y) = z = a*x + b*y + c
              f(x,y) = z = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2

Equating the two expressions for 'z', we have the equation of the line. Of course, this could be simplified algebraically by hand, but that is purely a matter of aesthetics:

     p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2 =  a*x + b*y + c

Best Answer

Related Solutions

MATLAB: How to fit custom polynomial equation

MATLAB: How to find the equation of the line that intersects a 3D surface and a plane passing through 3 points of interest on the surface

Related Question