Representation of a data set through a curve

Question

I have some doubts regarding the nomenclature given to the process of representing a data set through a curve (or function).

I have an exercise that asks me to make a parameterization of a set of exeprimental data (the data represents the cross section of nuclear reactions). In my view, this would be to find a curve that passes through the points, using a y(x) function in the 2D plane. I did this by dividing the domain into some parts and applying some polynomial interpolation techniques, so that for each region, I have a certain polynomial function that describes this data. Joining these functions, I have a curve that passes through all the points that represent the experimental data.

Here is my doubt, in this case, what I did was a parameterization of the data? An interpolation? Or a curve fit? Or these concepts are interchangeable in this case.

Thanks in advance!

Answer 1

If I may suggest, having $n$ data points $(x_i,y_i)$ along a curve, instead of using piecewise functions (your polynomials), use parametrization.

This means that you could build two cubic splines $x=f(i)$ and $y=g(i)$ which will go through all points $(i,x_i)$ and $(i,y_i)$.

Now, if you want to inerpolate, consider that $i$ is a continous variables.

This procedure is parametric splines (have a look here).

Answer 2

If you're fitting a function inside the bounds of your data, it's called interpolation. If you're fitting a function outside the bounds of your data, it's called extrapolation.