I would consider [spline curves][1], specifically cubic splines. They are smooth curves, can have fixed endpoints and can be deformed into each other relatively easily. Also there are many software libraries that implement them.

[This paper][2] includes an algorithm to fix the length of the Bezier Curve.


  [1]: https://en.wikipedia.org/wiki/Spline_(mathematics)
  [2]: https://pdfs.semanticscholar.org/b96a/4066a532713d31fcd4e54cfd4e0c1511a0aa.pdf