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You probably didn't. 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

You probably didn't. 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]. [0] Two-dimensional curve synthesis using linear curvature elements https://www.sciencedirect.com/science/article/abs/pii/0010448577900446 https://www.sciencedirect.com/science/article/abs/pii/001044...

raphlinus 6y ago
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.
- delhanty 6y ago
  Fair enough - aware that you're not a fan of biarcs from your thesis. >The simplicity of the arc as primitive curve has some nice features though, for example the fact that parallel curve is nearly trivial. Yes. Also, IIRC, Alexey Kurnosenko uses them in one of his papers to bound the regions of when it's possible to fit a spiral to end point data, https://arxiv.org/a/kurnosenko_a_1.html https://arxiv.org/a/kurnosenko_a_1.html
- delhanty 6y ago
  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.)
  - raphlinus 6y ago
    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.
    - delhanty 6y ago
      Thanks for those ideas.

raphlinus 6y ago

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.

delhanty 6y ago
Fair enough - aware that you're not a fan of biarcs from your thesis. >The simplicity of the arc as primitive curve has some nice features though, for example the fact that parallel curve is nearly trivial. Yes. Also, IIRC, Alexey Kurnosenko uses them in one of his papers to bound the regions of when it's possible to fit a spiral to end point data, https://arxiv.org/a/kurnosenko_a_1.html https://arxiv.org/a/kurnosenko_a_1.html
delhanty 6y ago
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.)
- raphlinus 6y ago
  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.
  - delhanty 6y ago
    Thanks for those ideas.