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I am looking for an interpolation between two points $P$ and $Q$. I need the curve to have derivative (direction) $\vec{v_1}$ at point P and $\vec{v_2}$ at point Q. In addition, there is a maximum curvature constraint (represented as minimum radius $r$). I want to find a curve that satisfies those constraints and having minimum distance.

Is there an analytical solution or does any programming language package (e.g. MATLAB) support numerical calculation for this?

If this one is hard to find, would dropping the minimum distance constraint and instead adding minimizing overall curvature be easier?

P.S. This spline may be a famous one but I don't know the name. Searching online by these criteria does not give me meaningful result.

Thanks.

zyl1024
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    This sounds like a very difficult problem in general. If you drop the curvature constraint, you can just use Hermite splines. But for the others...sounds tough! – John Hughes Nov 11 '14 at 22:06
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    Is this related? http://math.stackexchange.com/questions/689619/how-to-find-the-equation-of-one-curve-in-xy-plan-which-satisfies-such-functions – DanielV Nov 11 '14 at 22:47

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For one-dimensional interpolation the cubic spline is known to minimize the curvature (not exactly but "as measured by the second derivative"). So if you turn to minimizing overall curvature instead you may want to use http://en.wikipedia.org/wiki/Cubic_Hermite_spline. It is probably the most famous one.

Community
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rych
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  • The curvature of a parametric curve is not identical to its second derivative. (Also the cubic spline minimizes the total squared second derivative, not the maximum second derivative.) –  Jan 05 '15 at 05:51
  • Thank you, @Rahul. I've edited accordingly. I often don't know how much detail to include or omit. If you're satisfied could you upvote back. – rych Jan 05 '15 at 06:20
  • I still don't think it's particularly useful because the question explicitly talks about geometrical curvature, "represented as minimum radius $r$", and a cubic spline with small second derivatives can still be arbitrarily bad in that respect. But I've retracted my downvote. –  Jan 05 '15 at 11:32