Inter-, extra- and intra- polation

Interpolation: the mathematical problem of interpolation is to find a function that goes exactly through the training points. It doesn't say anything on the inputs on which this function will be later evaluated. Of course, the solution provides a way of evaluating the function in unseen inputs, but it really is not about that, it is about "going through the given points".

Extrapolation: is about the relation between the new inputs where the function will be evaluated and the inputs used for training. It doesn't say anything about the relation between the function and the training inputs, as interpolation does: you can extrapolate using an interpolating function (i.e. that goes exactly through the training inputs) or an smoothing/approximating function (i.e. that goes close to the training inputs). Extrapolation means that the new inputs are "outside the region delimited by the examples we used for training (the observation range)", for example one could say that extrapolation is evaluating a learned function outside the convex-hull of the inputs used for training (this example applies only if the input set has a notion of "inside" and "outside", which is the case in many many situations). In many dimensions it could be difficult to define what is inside the observed region, and surely there are may ways fo doing it.

Intrapolation: is a term I coined (I am sure I am not the first! do you know if anybody used it first? or maybe another clever word to express the same idea?). As in extrapolation it is about the relation the new inputs have with the inputs used for training. It doesn't say anything about the function we are using, as interpolation does. You can either intrapolate with an interpolating function or with an smoothing/approximating function. Intrapolation means that the new inputs on which the function will be evaluated are within the observation range, following the previous example, they would be inside the convex-hull of the inputs used for training.

Hence, the complement of extrapolation would be intrapolation, whether we are using an interpolant or not. I think this makes the jargon cleaner!



Summary:
interpolation ≠ smoothing/approximation
intrapolation ≠ extrapolation