Partition of function into pieces for interpolation needs

Question

I've got some experimental data obtained from my mate's research. There are two sets of (x,y) points for each curve. He asked me to interpolate function values between this points, so for each curve I implemented Akima's interpolation method and got result shown below.
As you can see dots were taken irregularly for each curve and between them as well.
I can easily evaluate y(x) values for each curve, but the most valuable thing is to interpolate values between curves (curves illustrate dependance of surface from parameter, so each curve correspondents defined parameter).
I divided each curve on segments between know points and evaluated cubic polynomial coefficients a,b,c,d. Now I would like to interpolate coefficients of polynomial on each segment, but the problem is that experimental dots are irregular, so segments doesn't locate under each other.
So, I would like to ask your suggestions how to "synchronize" segments.
Actually, I don't understand how can I connect, for example, grey dots (curve2) with curve1: should I draw perpendicular and it will be enough or not?

Answer 1

If you have a functional representation of two curves with parameter values $t=0$ and $t=1$... then for the curves in between you could use linear interpolation $$(1-t)f_0(x)+tf_1(x).$$

If you can discretize the two functions on the same set of nodes $x_1<x_2<\ldots<x_n$, then the interpolation is done on the column-vectors, $$\big[f\big]=\begin{bmatrix}f(x_1)\\\vdots\\f(x_n)\end{bmatrix}=(1-t)\big[f_0\big]+t\big[f_1\big].$$