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!!!
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