Sunday, April 11, 2021

[rgawvokb] minimal smooth curves

given a set of points, find the shortest smooth path through them (traveling salesman).  "smooth path" remains to be precisely defined.  perhaps infinitely differentiable everywhere (but that might need a more careful definition).  perhaps we also want analytic.  (wikipedia: smooth but not analytic, see also articles that link to it.)

also need to bound curvature.  is bounding curvature the best way to do it?  is that all that is necessary?  is it OK for higher derivatives to be arbitrarily large?

splines are the typical solution for this kind of problem, but we avoid them because we want infinite smoothness.  but not sure what this is useful for.  naively constructing a polynomial between points results in crazy behavior: goal here is to tame it.

simpler problem: construct a polynomial through points in order left to right, staying within given bounds on its 0th, 1st, (or more) derivatives.  1st derivative (slope) bound could be a given delta beyond the slope of the line segment between consecutive points.

maybe slope must be zero at extremal given points.

polynomial through points on the complex plane.  bound magnitude?  real part?

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