Confused about the true meaning of OnCell and OnGrid

[timholy]

This may sound surprising, but I'm looking for clarification about what OnCell and OnGrid actually mean. Let's take the case of Constant: my interpretation is that OnCell should mean that the array value is constant over i-0.5 to i+0.5 but has a step discontinuity at the edges of this region. Conversely, I would expect OnGrid to be constant over i to i+(1-eps()). In other words, something like this:

But that's not what we have, and indeed OnGrid and OnCell seem to behave identically:

Likewise for Quadratic:

By the definition I am picturing, I'm not even sure there is an obvious meaning to Linear, OnCell that is also continuous. (The closest I can come up with is that the slope at i is (y[i+1]-y[i-1])/2, but that has discontinuities at the half-integers.)

So despite having worked with the package a lot (though not for quite a while, truth be told), I am confused.

[tomasaschan]

I think the original discussion of OnGrid vs OnCell arose out of needs for handling boundaries on or between the given data points (see e.g. the discussion in #1, as well as #9 (comment) with following comments), with the intent of having the same results on the interior. Thus, I'm not sure it's supposed to make a difference except at the edges.

This realization might make it easier to find better names for the concepts.

[tomasaschan]

Re: what it would mean to have Linear, OnCell in the interior, I recall drawing a graph with the knots between the data points, which basically looked like an optimization problem: find the series of slopes that create a continuous graph, with as small slope-jumps as possible. I think I also found a way to formulate that problem in terms of linear B-spline coefficients, but I have no idea where I found that, or if I documented it at the time. I think we discarded the concept because it was really silly and we couldn't imagine anyone ever preferring that to "normal" linear interpolation.

[timholy]

Thanks for the reminder!

OK, if this really is just about the boundary conditions, I might think about moving this to be more closely attached to the BC object. But not the highest priority.