I was wondering if I could get some input on my strategy for inverting a set of multivariate polynomial equations for the independent variables. My functions are as follows: $$g_1(x,y) = a_0 + a_{1}x+a_{2}y + a_{3}xy + a_{4}y^2 \\ g_2(x,y) = b_0 + b_{1}x+b_{2}y + b_{3}xy + b_{4}y^2 $$
Now, what I want to get are analytical expressions for the variables x and y expressed as a function of the independent variables. In other words, I want to invert my system of multivariable functions and derive the dependent variables as functions of the independent variables. $$x= f(g_1,g_2) \\ y = f(g_1,g_2)$$ The two surfaces $g_1(x,y)$ and $g_2(x,y)$ do have an intersection, below is the MATLAB plot of the two:
I was wondering if as a strategy, I would solve the line of intersection between the two surfaces to get y as a function of x: $$g_1(x,y) = g_2(x,y) \Rightarrow y = f(x) $$
And then I could put this into my initial system of equations to change the problem into a single variable polynomial problem:
$$g_1(x,y) = a_0 + a_{1}x+a_{2}f(x) + a_{3}xf(x) + a_{4}f^2(x) \\ g_2(x,y) = b_0 + b_{1}x+b_{2}f(x) + b_{3}xf(x) + b_{4}f^2(x) $$
And then I could solve this set of polynomial equations to go from $$\begin{bmatrix} g_1(x,y) \\ g_2(x,y)\end{bmatrix} \Rightarrow \begin{bmatrix} x(g_1,g_2) \\ y(g_1,g_2)\end{bmatrix}$$
I wrote a badly phrased question yesterday - apologies for that. Would this strategy work at all?