Smooth arclength interpolation

[SouthEndMusic]

Over the past few weeks, in my hobby math and coding time I have been fixated on a question that arose here, namely: What is a good way to interpolate through a set of points in $\mathbb{R}^n, n \ge 2$ , which is $C^1$ smooth and parameterized by arc length?

I understood that a common approach here is to make an interpolation that is not by arc-length and then approximate the arc-length parameterization using numerical methods. I however figured there should be a way to do this without such heavy lifting. I tried several algebraic approaches which didn't really get me anywhere, so then I started looking for a more geometrically informed approach.

A quick Google search reveals this pathological method, which suffers from not being at all flexible, the effect of points being non-local, and having arbitrarily large detours if the points are distributed in an unfortunate way.

The core idea I had is to again start with a non arc-length parameterized curve, but then to approximate it with segments that are trivially parameterizable by arc-length, namely line- and circle-segments. I explain the full construction of the method in a notebook that I checked into this branch.

I am not sure this method is suitable for DataInterpolations.jl, but it can make good use of the internals, so maybe it should be a separate package depending on DataInterpolations.jl.

I now use QuadraticSpline as the default 'shape-defining' curve, because it is a method that is $C^1$ smooth and supports vector valued interpolation. However, for this method also the effect of data points is non-local, which was my motivation for opening #401.

I am btw not completely sure what to put in the docstring of struct SmoothArcLengthInterpolation, because there are several different methods which are different entrypoints in the construction of the interpolation.

[yolhan83]

Wow, you really went through that 👍 for usefulness, for now I only really got the case of advection simulation on the curve where you really need that, I don't know if there are others.

[ChrisRackauckas]

Looks reasonable to me.

I am btw not completely sure what to put in the docstring of struct SmoothArcLengthInterpolation, because there are several different methods which are different entrypoints in the construction of the interpolation.

🤷 put the notebook up somewhere and cite that.

But the notebook shouldn't be in the history of the package, since that would cause repo bloat.

[ChrisRackauckas]

Requires a squash merge to make sure the memory of the notebook isn't in the core repo.

[ChrisRackauckas]

Just realized, this is missing tests.

[SouthEndMusic]

Yep, I'm on it