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I have a two dimensional plane with $U, V$ coordinates from $0,0$ to $1.0,1.0$.

The $U$ coordinate is projected to $\mathbb{R}^3$ and primarily contributes to an $X$ coordinate. However, the rate of change along $U$ is not uniform to the changes in $X$. The same is true for the $V$ coordinate in the plane and it's relation to the $Y$ coordinate in my $3D$ space.

More concretely I have a $2D$ plane of $UV$ coordinates and I am calculating the way light refracts through an object. In the simplest case the refraction indices of my two mediums would be equal, and all the refraction vectors for each $UV$ coordinate would simply be $0,0,-1$, or straight lines through the plane. Essentially what you see when you look out a thing glass window.

In a more complicated case, like with a sphere of some other material the refraction vectors differ. I can calculate my data points but I would like to derive a function of the following form:

$$f(u,v) = \overrightarrow{V_r} = (x,y,z)$$

Assuming I have a non-uniform set of values for $x, y$, and $z$ in relation to the $(u,v)$ coordinates how can I go about discovering the function to describe the $x,y,z$ values.

I hope this is clear, let me know if other details are necessary.


Edit: Let me try a different likely more meaningful explanation.

I have a $2$ dimensional surface. Each point on the surface corresponds to a $3$ dimensional vector. The surface is square and has length and width equal to $1$; though this point might actually be irrelevant. Let's start with a simple case, in which the surface points mapped to the $3D$ vectors are symmetric along both the $X$ and $Y$ axes. I don't have the function that maps my $2D$ surface point to the corresponding $3D$ vectors, that is what I want to find, but in a simple case the mapping is 1:1. I can calculate, through a series of long, complicated and resource heavy calculations the $3D$ vector for any point in my $2D$ surface. I have done that for $1024*1024\approx1,000,000$ points. So I have a million $3D$ vectors. How can I go about deriving an approximation function that will give me the $3D$ vector given some $2D$ point in my surface.

Image I have the surface (not the one shown in the image), I want to derive the function of the surface in terms of the horizontal and axial axes. As opposed to the image, the Horizontal and axial values may not equal the "$X$,$Y$" coordinates of my surface. That is I am not only looking for "height", that we'll call the $Z$ axis.

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Hi, I'm not sure what you mean by coordinates of a plane, are they the coordinates of the generating vectors, or all points on the plane? Also your objective is not clear, do you want to derive a refraction law? I think it would be helpful to rewrite the question, adding some definitions and clear mathematical statements. –  Artur Gower Sep 10 '13 at 13:38
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