Ideas for 2D -> 3D representation

[yolhan83]

The 2D model allows for a 3D parametric curve c(s), the curvature is then called within the source-term. Will expose ideas when they come here.
Q:

1. Should the curvature stay a function or become a variable even though we don't need it's derivative and we don't need it in the flux?
2. Should someone only define a curve and the curvature could be implied ? Deps to ForwardDiff would be added.

[SouthEndMusic]

If you define the curve(s) with DataInterpolations.jl (or something similar), you get derivatives for free without having to depend on an AD backend

[yolhan83]

Yes but I should support explicit paramétrisation too no ? Also I may just let all that to the user and require the functions curve(s), er(theta,s) (or normal(s) and binormal(s) and curvature(s) that would be the most general. I will code that and then I will think about those helper.

[yolhan83]

Also I would need to force the paramétrisation to be arc-length parametrised and make sure it goes well when the normal is zero.

[yolhan83]

For now it will look like this : (Did not touch the curvature so its set to 1 which allow to see angular variations but it should be a lot lower, to edit it simply do BloodFlowTrixi.curvature(s) = scalar value (around pi/40 here) )

∧(v1,v2) = SA[v1[2]*v2[3]-v1[3]*v2[2],-v1[1]*v2[3]+v1[3]*v2[1],v1[1]*v2[2]-v1[2]*v2[1]]
curve(s) = SA[s/40,cospi(s/40),sinpi(s/40)]/sqrt(1/40^2+(pi/40)^2)
tanj(s) = SA[1/40,-pi/40*sinpi(s/40),pi/40*cospi(s/40)]/sqrt(1/40^2+(pi/40)^2)
nor(s) = SA[0,-cospi(s/40),-sinpi(s/40)]
binor(s) = ∧(tanj(s),nor(s))
er(theta,s) = cos(theta)*nor(s) + sin(theta)*binor(s)
for i in eachindex(sol)
    datas = get3DData(eq,curve,er,semi,sol,i,vtk=true,out=out)
end

[yolhan83]

for interpolation I need some kind of mix between BSpline and Hemite to get a curve with its tengent norm equal to 1 at every point. Did not find anything like that in Interpolations.jl and Dierckx.jl. DataInterpolation seems to not support parametric curves meaning it will support only curve with shape (s,c(s),0) and no chance of being arc-length here (would mean c'(s) = 0 )

[SouthEndMusic]

This is a fun problem: finding a $C^1$ smooth interpolation by arclength through a given set of points (or at least 'informed by' these points e.g. a spline curve).

DataInterpolation seems to not support parametric curves meaning it will support only curve with shape (s,c(s),0)

Note that DataInterpolations does support parametric curves in higher dimensions (i.e. $\mathbb{R} \rightarrow \mathbb{R}^n$), which can be non-allocating if you use SVectors.

I could add an interpolation type to DataInterpolations.jl if we agree on something useful (I am familiar with the internals of DataInterpolations.jl but not your package), although this is a bit removed from the idea of DataInterpolations.jl so it might need a different/new package.

I found a suggestion here, but that one seems quite restrictive / hard to handle in practice. This paper seemed promising at first but I don't think it's actually what we want.

[yolhan83]

Yeah, it may be too specific, but the paper seems interesting. It's trying to minimize the length of the curve, though I'm not sure it will help here. Maybe a method to add a constraint to a spline problem is general enough to be useful if it can be written with Optim.jl.
I would just minimize ( norm(c'(s))-1 )^2 in this case?
Or just a way to impose derivative norm on those points is already quite general and may be usefull.
Note that I have never touched spline so I may do little sentence that leads to insane problem sorry :')
I will take a closer look at DataInterpolation and if a more precise idea comes I will open an issue there.

[SouthEndMusic]

You could use that the derivative of the curve makes a continuous path over the unit sphere parameterized by 2 angles. Maybe make a linear interpolation trough the angle space and optimize for reaching the data points (possibly putting a bound on how fast the angles can change). Although this is getting quite complex

[SouthEndMusic]

I think the best way to approach this if you are solving an implicit problem anyway is to use an existing non-arclength interpolation and solve for the arclength reparametrization

[yolhan83]

Yes, with enough points that won't be a problem, its basicaly finding a squeleton at this point