Modelling ellipsoid-like surface with equations In my numerical simulation, I would often have to generate the failure surface of some materials. Often, the failure surface will take a ellipsoid-like ball such as this:

Of course, the surface will vary depending on the material properties and other parameters, however, generally it will take the form of ellipsoid-like surface like the above.
Now, to determine the surface, I am thinking about determining points on the surface, and then use interpolation techniques to connect those points together to form the surface equations. The reason for this is simple, it is completely inpractical and somewhat outright impossible to generate the surface from the physical first principles. So an approximation is needed here. In other words, what I want to do is something like a higher-dimension of power series. For power series, if you determine enough points, you can fill in the coefficients and then you can form the power series plot and draw on a graph paper, a nice plot of x vs. y. I am planning to do the same for the surface plot in this case. And of course, the more the points are, the more accurate the series/surface is. 
Anyone knows what are the general equations ( with coefficients) that I can use to approximate in this case? Or are they more efficient techniques that I can use?
 A: I'm not sure I'd call the image you depicted “ellipsoid-like”, but the generalization of a power series is easy: choose the degree $n$ of your surface a priori. Then you are looking for an algebraic manifold with the equation
$$ \sum_{i=0}^n \sum_{j=0}^{n-i} \sum_{k=0}^{n-i-j} a_{ijk}\,x^iy^jz^k = 0$$
This is an equation with $\binom{n+3}{3}$ coefficients $a_{ijk}$ which describe your surface. In the case of a quadric surface ($n=2$), which would also cover real ellipsoids, you'd get
$$ a_{000}
+a_{001}z
+a_{002}z^2
+a_{010}y
+a_{011}yz
+a_{020}y^2
+a_{100}x
+a_{101}xz
+a_{110}xy
+a_{200}x^2
= 0 $$
This equation is homogenous, i.e. multiplying all the coefficients by the same constant will result in the same surface. So you can choose any one of them and simply assume it to be $1$. If you are very unlucky, that coefficient would have to be zero in the actual solution, in which case the other coefficients would tend to infinity as you try to determine them. But perhaps you can identify a known non-zero coefficient from your application up front.
Now if you have $\binom{n+3}3-1$ data points, you can write down as many equations of the above form, and exactly solve them for the coefficients $a_{ijk}$. This is interpolation. If you have more data points, then you can do a least squares approximation of the coefficients. That's an approximation of your data points by the best fitting algebraic surface of the given degree.
