3 ms·
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
by 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 agoI 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 agoThanks for those ideas.