Suppose we have been given some closed curve, described by $f(x,y)$. Is there any way to figure out, what are the directions of the principle axes?
I know how to do it if $f(x,y)$ is a degree-$2$ polynomial. The two methods I've seen, do the some in the following way :
(i) Consider $f(x,y)$ to be a quadratic form, and then write it in matrix form. Then find the eigenvectors, that diagonalize the matrix. These eigenvectors are the directions of the principal axes. I believe this is the same as writing the hessian matrix for the curve and then finding the eigenvectors.
(ii)Consider a line $y=mx$. Substitute this into the equation and find an expression for $x^2$. If the curve is centered about the origin, the square of the distance of any point on the curve, to the origin would be $x^2+y^2 = x^2(1+m^2)$ Plug the expression for $x^2$ into this, and we'll get a function $g(m)$. Then we find the maxima of $g(m)$ w.r.t m, by setting $\frac{ d g(m)}{dm} = 0$.
We'd obtain values of 'm'. Plugging this back into $y=mx$, we get our principle axes vectors. I don't understand the exact intuition behind this, if someone can explain, that would be helpful. I've seen someone do this in a question, I'm attaching a link to it : Question.
In this link, you see the two above methods applied to solve a problem. However, what if the curve was of a higher degree. How do we solve the problem?
For example, say $f(x) = x^4+9y^2-10xy=100$. How do I find the directions of the principal axes for this curve? Using the second method as described above, I obtain 4 values of $m$. These are $m=2.14612,-2.63542,\infty, \frac{5-\sqrt{205}}{9} $ Hence I'm obtaining 4 different axes. Which ones if any of these, are the principal axes?
Or is there any other way to solve these kinds of problems. Writing a curve in matrix form and finding eigenvectors, seems to be the easiest, and most intuitive, but how do I write a polynomial of degree higher than $2$ in matrix form ?
Here is a graph of the function and the supposed axes, that I obtained. graph
By principal axes, I'm refering to the axes about which the tensor of Inertia has only diagonal terms.