# How to find a surface from two lines?

sorry if this is a basic problem but I don't know where to start looking.

Imagine two perpendicular lines ("profiles") in a "$T$" spatial arrangement. The lines are arbitrary (empirical functions should I say?) in the sense that they don't follow any simple formula (but I have the data of each line, for example they could be height of terrain along a 10 km transect, with data every 500 m). One line is parallel to the $X$ axis, the other is parallel to the $Y$ axis. These lines have one point in common, ie there is one value where $Xi=Yi$.

I would like to interpolate between them to guess how the area would look like. So I want to go from 2 known lines (1D) to a surface (2D).

I imagine that I need a function $Z=f(X,Y)$ such as when plotting this function I can have a 2D representation of the surface containing both lines ("profiles")

For simplicity, it is OK if: - The interpolation is the simplest (linear?) - The lines are perpendicular (but it would be nice to have a solution for any given angle)

I am sure this must be a VERY common problem in many many fields... But I don't know the math world.. Could you please provide some keywords so I start looking??

Thanks very very much!!!

Edit (25-7-2013):

I'm not sure if I explained myself correctly, so I made a couple of plots to illustrate my points. The lines are NOT straight lines (except when viewed from above). Imagine you measure the height of the terrain over two lines: one from A to B, another from C to D. When viewed from above, these lines are perpendicular and they have a point in common: they look like this: https://www.dropbox.com/s/z92j3ae417kp65u/1d.lines.png

Note that the point they have in common is the beginning of one line, and the middle of the 2nd line.

Now, what I need is an algorithm which allows me to "guess" the surface defined by these two lines. I would think on a simple interpolation, but how? I need to obtain this: https://www.dropbox.com/s/3is8zzetps4coq3/surface.png

Your lines must intersect to determine a plane; skew lines don't define a plane. But if they do intersect, then produce 3 non-collinear points $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$, $(x_3,y_3,z_3)$. In general, you will have a planar equation of the form $z = ax + by + c$ (except for the special case of a vertical plane). Your coefficients are the 3 unknowns, and you have three points. Using these 3 points, you can solve for your coefficients and get an equation for your plane.