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I am curious if a spline can be both relaxed (second derivative = 0 at both endpoints) and clamped (first derivative is explicitly defined at both endpoints). This only needs to be true for a single spline between two end points. If this is not possible, what power (quartic, quintic, etc.) of spline would be required to fulfill these conditions?

Either way, how would I go about calculating a spline that fulfills these conditions?

I know very little about splines, so I may be missing something obvious. Any help or additional resources would be great. Thanks!

nardavin
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  • Did you find a solution to this? I am looking for a way to do a cubic spline interpolation between points, but have pre-defined first order derivatives and 0 second order derivative at the two ends. Thanks – Confounded May 15 '19 at 17:51

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Since you have 6 equations (2 points, 2 first derivatives and 2 second derivatives) to satisfy, the interpolating spline will have 6 control points as well. This means that the spline could be a quintic Bezier curve, a quartic b-spline curve of 2 segments or a cubic b-spline curve of 3 segments.

fang
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  • I'm sorry, but I have very little experience with splines. Why would I need anything with more than one segment to accomplish this? Also, if I understand what you are saying, I would have to (effectively) use a quintic curve to solve this. How would I begin to calculate a spline to satisfy the conditions? Feel free to just leave some resources for me if you believe that would better answer my question. I have searched but can't find anything of use. – nardavin Nov 28 '16 at 05:15
  • If a quintic spline is good for you, you can use "quintic hermite" curve which is defined by 2 points, 2 first derivatives and 2 second derivatives. You can find out how to convert a quintic hermite curve to a quintic Bezier curve on the web. – fang Nov 28 '16 at 15:17
  • How would I go about finding the equation for a quintic hermite spline? I can't seem to find the equations to calculate the coefficients based on these values anywhere. I have tried to derive them by hand, but I keep getting errors. – nardavin Nov 28 '16 at 19:11
  • Try this link: https://www.google.com.tw/url?sa=t&source=web&rct=j&url=https://www.rose-hulman.edu/~finn/CCLI/Notes/day09.pdf&ved=0ahUKEwjMpaeQ1czQAhWCWbwKHWu4CrQQFggYMAA&usg=AFQjCNFCn2VLFotyRINyQ0aMWYY9PbtFgg – fang Nov 29 '16 at 00:05
  • Aha! Thank you very much. – nardavin Nov 29 '16 at 00:53