If this paper is interesting to you, you may also be interested in this amazingly well done video on the continuation of splines. It isn't exactly the same thing, and delves into some deeper concepts, but I've watched it like 3 times over the last year.

https://www.youtube.com/watch?v=jvPPXbo87ds

Interestingly, I've never heard about cosine interpolation until now. I wonder why? Could it be that in a practical computation, cosine is in fact a polynomial itself and cosine interpolation is a kind of polynomial interpolation with extra steps? Does this also mean that we can "tame" approximating polynomials of high degree by introducing even more degrees?

Fascinating.

It's a trick. His example for cosine interpolation is misleading in the sense that the sequence of points' heights is oscillating.

The cosine interpolation simply ensures that the interpolated-function derivative is 0 at the control points. If anywhere in the sequence there was a point that is lower / higher than both its predecessors the downsides of this technique would become obvious.

Yeah, I was surprised the example input didn't have a sequence of 3 monotonic points to reveal this.