Surface interpolation from two lines

Question

Sorry if this is a basic problem but I don't know where to start looking (mainly because being an outsider I don't know the terms and nomenclature).

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. Each line is a velocity profile along a transect. 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 a point i where Xi=Yi.

I would like to interpolate between them to guess how the velocity distribution would look in the area defined by these two lines (eg the containing rectangle?). 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")

The lines are perpendicular. The lines are NOT straight lines (except when viewed from above). Imagine you measure velocity 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/1ju2hzmkvijlz2j/1d.lines.vel.jpg

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" (2D velocity vector field) 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

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!!!

Answer 1

Interpolation refers to finding data between points. In other words, if you had the data on two perpendicular line segments, all you could do with interpolation is fill in something in the triangle enclosed by the two line segments, but not in an entire rectangle. For example, you could use a linear interpolation of the values between points $(i \Delta x, 0)$ and $(0, i \Delta y)$ at which you know the values.

For the rest of your rectangle, you can't do inter polation but you could do extra polation.

Basically, however, extrapolation has a lot to do with what you think you know the real data looks like. Should it be smooth? Should it be oscillatory? Whatever assumptions you put in, you will get out.