polynomial function 3D

Hello,

I would like to put a second order polynomial function through 9 points. The points are given in the form P(x,y,z).

Goal:

(1) Calculate polynomial function

(2) Calculate the mean curvature (= second derivative) of the function

Is there something similar to "p = polyfit(x,y,z,n)"?

Remark:

I can put a circle through three points in space and calculate the corresponding curvature of the circle. For this I use "circlefit3d(p1,p2,p3)". But I don't want to use only 3 of the 9 points for the curvature calculation.

I would be very pleased about a feedback.

[Bjorn Gustavsson]

Question to you: Is the curvature constant along a polynomial?

[Christoph Thorwartl]

3D markers were fixed on the surface of an object (in a row). A force is applied to the object which causes it to bend. The bending line does not have a completely homogeneous curvature.

[Christoph Thorwartl]

Picture see attached.

Answer 1

[Bjorn Gustavsson]

You have the file exchange contribution polyfitn that should do what you want.

HTH

Answer 2

[Walter Roberson]

Is there something similar to "p = polyfit(x,y,z,n)"?

No. However, polyfit() constructs the Vandermode companion matrix and then uses \ for linear least squared computation of the coefficients. You can do the same thing;

X = x(:); Y = Y(:); Z = Z(:);
A = [X.^2, X.*Y, Y.^2, X, Y, ones(size(X))];
b = Z;
A\b

Assuming a*x^2 + b*x*y + c*y^2 + d*x + e*y + f = z .

In particular, assuming that there is no z^2 or x*z or y*z .

If there are higher order z, then at the moment I am not sure how to set up the problem without ending up with a companion matrix and all zeros on the right hand side, which unfortunately leads you to an all-zero solution instead of something useful.

[Christoph Thorwartl]

Thank you very much for your answer. I will now look at both approaches in detail.