find the equation of a 'level' curve, for a function $\mu(x_1,x_2)=0$

Question

I would like to ask the following question.

I have an aggregated function, $$ \mu(x_1,x_2) = \|\nabla f_2\| f_1(\vec{x}) + \|\nabla f_1\| f_2(\vec{x}), $$

where the norm of the gradients are also functions of $x_1$ and $x_2$.

I would like to get the equation describing the curve for which $\mu = 0$.

As an example, for the functions,

$$ f_1 = (x_1 - 3)^2 + (x_2 -3)^2 $$ and $$ f_2 = \frac{1}{2}(x_1 - 1)^2 + (x_2 - 1)^2 $$

I get,

$$ \mu(x_1,x_2) = \left[ (x_1 - 1)^2 + 4(x_2 - 1)^2 \right]^{\frac{1}{2}} \left[ (x_1 - 3)^2 + (x_2 -3)^2 \right] + 2\left[ ((x_1 - 3)^2 + (x_2 - 3)^2\right]^{\frac{1}{2}} \left[\frac{1}{2}(x_1 - 1)^2 + (x_2 - 1)^2\right] = 0, $$

and I need some function describing the solution, like $\gamma(t)$ or $\gamma(x_1,x_2)$. And it should be goning from $(1,1)$ to $(3,3)$. It would be nice to get a general approach explained, as I want to do this for other more complex functions and with more dimensions. However, I would be happy with the solution to this example since I use it in the report.

(I'am not sure if it is possible or easy, I'am just at a lost and any comment helps.)

Answer 1

Considering $f_k =\frac 12 (p-p_k)^{\dagger}\cdot A_k\cdot(p-p_k)$ with $A_k > 0$ then $\nabla f_k = A_k\cdot(p-p_k)$ and $||\nabla f_k(p)|| > 0$ for $p \ne p_k$

Resuming,

$$ ||\nabla f_1(p)||f_2(p)+||\nabla f_2(p)||f_1(p) > 0\ \ \mbox{for }p_1 \ne p_2 $$

so in those conditions, the solutions for

$$ \mu(p) = ||\nabla f_1(p)||f_2(p)+||\nabla f_2(p)||f_1(p) = 0 $$

are at $p = \{p_1, p_2\}$

Here $p = (x_1, x_2)$

Attached a level curve plot for $\mu > 0$. In red $p_1, p_2$