For the Prior Work, I gave you this reference on Twitter in response to your tweeted question.
>Computational Geometry: Curve and Surface Modeling by Liu Ding-Yuan and Su Buqing - Chapter 2 covers Béziers. Originally published in Chinese in 1989 - Chinese shipbuilding industry splines. Maybe of interest? (IIRC you don't reference it in your thesis.)
Thanks for the reference. I had a quick look now, and I wish I knew about this when I wrote my thesis. For the most part, it seems to cover the standard spline literature (Farin, de Boor, etc), though it goes into more detail than usual about the nonlinear splines.
I didn't find anything in the book about this curve fitting problem specifically, though it's possible I missed it.
You seem a concientious author and I thought that, from having read your thesis, you were probably not aware of the book and might like to know about it.
There's interesting material in there on nonlinear splines because its a fairly old book (1989) that summarizes how they used nonlinear splines, biarcs etc in the Chinese shipbuilding industry in the decade or two priot to that.
Particularly, the survey of methods on how to choose free parameter in the biarc is not something you see much these days.
Edit: Interestingly, like you they refer to the "Euler Spiral - rather than "Cornu Spiral" or "Klothid"/"Clothoid". Another name, I see in literature from that era is "LINCE" (Linear Curavture Element) such as in [0].
I'm definitely familiar with biarcs (and did talk about them a little in the thesis) because that's the basis of Ikarus. But generally I'm not a fan of biarcs, I feel Euler spirals can do pretty much anything biarcs can, just more smoothly. The simplicity of the arc as primitive curve has some nice features though, for example the fact that parallel curve is nearly trivial.
Shipbuilding has a long connection with splines, long before digital computation. The Autokon system was built by Even Mehlum in the 60s for the Norwegian shipbuilding industry, and this was a pioneering application of the Euler spiral spline. It's not surprising to learn the Chinese shipbuilding industry was also doing interesting things and engaging with mathematicians such as Liu Ding-Yuan and Su Buqing.
IIRC, in your thesis, the method of approximating Euler spirals is exact at the midpoint and has errors at the end.
Say if instead, one wanted the reverse - an Euler spiral approximation that was exact at the endpoints and was allowed errors in the middle. Do you have any idea of the best method to achieve that?
(So to make it concrete, if one had end points and end tangents that were compatible with fitting a spiral, which should define a unique Euler spiral segment. Let's say it's parameterized by turning angle rather than length.)
I think I understand, you're talking about the numerical integration, right? It's probably something of an academic question because you can just set the error to 1e-9 or whatever, and the convergence of that approximation is so good it doesn't slow you down appreciably.
That said, there are other things you can do. Since the derivatives of an Euler spiral are easy enough to compute analytically, you could do Hermite interpolation with a polynomial of arbitrarily high order. I've worked out how to do it for quintic, which is pretty tractable, but I'm sure it could be done even higher. That definitely meets your stated goal (precise at endpoints, sloppy in the middle), but the order of convergence of a quintic is only moderately good compared to the higher order polynomials in spiro.
Actually, they do give a comprehensive analysis of infection points and singularities for the cubic polynomial segment with given endpoints, tangents and magnitudes. There's a diagram of square with some (maybe superficial) similarity to the one in "High Accuracy Geometric Hermite Interpolation".
Chapter III Parametric Cubic Spline Curves
4 A Theorem Concerning Segments of Parametric Cubic Curves
Also, it's based on earlier papers by Su Buqing from around 1977.
>Computational Geometry: Curve and Surface Modeling by Liu Ding-Yuan and Su Buqing - Chapter 2 covers Béziers. Originally published in Chinese in 1989 - Chinese shipbuilding industry splines. Maybe of interest? (IIRC you don't reference it in your thesis.)
https://twitter.com/delhanty/status/1370618402654461953
In chapter 4, they have a construction for a GC2 Bézier spline space curves - pairs of quadratics and cubics.
Generally, they cover a lot of the same material that you do in your thesis.