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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?

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Just a fleeting though, perhaps this could be considered as the (possibly overdetermined) method of lines solution of a PDE, and then one could run an inverse method approach to reconstruct the original surface. –  Arkamis Nov 2 '12 at 16:03
    
The equations in the sets are determined experimentally themselves and don't represent differential equations, if that helps. –  The Dude Nov 2 '12 at 16:09
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1 Answer

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.

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Yes, thank you for the formalism. Is there a way to restrict the family of functions I need to fit by looking at the form of these maps? –  The Dude Nov 2 '12 at 16:28
    
@The Dude: If the given input consists only of the values $z_{kj}=\phi(x_k,y_j)$ then the method of choice is called spline interpolation. Look it up in a suitable text book, e.g., Gerald Farin: Curves and surfaces for computer aided geometric design; Academic press 1988. I'm sure there are newer books on this topic available, but that's the book on my shelf. –  Christian Blatter Nov 2 '12 at 20:51
    
Okay. I'll give it a shot. I've used b-splines for 2D fitting before, but this is my first real attempt at 3D fitting. Thanks! –  The Dude Nov 2 '12 at 22:00
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