Creating a 3D surface from 2D graphs So I have two sets of equations:
$\mathcal{A}$ = \begin{equation} 
    \{ f(y_{0},x), \, f(y_{1},x) , \;... \;, f(y_{n},x) \}
    \end{equation}
$\mathcal{B}$ = \begin{equation} 
    \{ g(y,x_{0}), \, g(y,x_{1}) , \;... \;, g(y,x_{n}) \}
    \end{equation}
And I created a surface out of these equations in a plot. I know I can find an equation for this surface by fitting the plot data to known surfaces, but that is not what I want.
I want to be able to directly derive an equation for the surface from just these equations. Is this possible? How can I do this?
 A: You are not given "equations", but $2n+2$ curves in three-space, namely the curves
$$\gamma_j:\quad x\mapsto\bigl(x,y_j,f(x,y_j)\bigr)\qquad(0\leq j\leq n)$$
in the vertical planes $y=y_j$, and the curves
$$\gamma_k'\quad y\mapsto\bigl(x_,y,g(x_k,y)\bigr)\qquad(0\leq k\leq n)$$
in the vertical planes $x=x_k$. These $2n+2$ curves form a wire mesh over the $(x,y)$-plane which from above looks like an orthogonal grid. Now you are in search of a function $$\phi: \ (x,y)\mapsto z=\phi(x,y)\ ,$$ such that the graph of $\phi$ (a surface over the $(x,y)$-plane) contains all these wires $\gamma_j$ and $\gamma_k'$.
In the first place such a $\phi$ is not uniquely determined at all. There are various principles to choose an "optimal" $\phi$ corresponding to the given data. Which of these principles suits best in your situation has to be judged by an "expert". In any case the problem formulated here is a problem of utmost interest in practical applications (e.g., sculpting of car fenders), and there is a large variety of numerical techniques (and software) available to deal with it.
