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Let me ask a question , given 2 points on the XY plane and given the 2 tangents at them, how to compute an arbitrary chosen smooth curve passing the 2 given points. For details, traveling along the curve in one direction, the curvature must satisfy (Condition A or Condition B) given

Condition A. the curvature does not increase ; Condition B. the curvature does not decrease.

Thank you in advance.

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    The special case of curvature changing *linearly* with length along the curve is called an [Euler spiral](https://en.wikipedia.org/wiki/Euler_spiral). You might find some information about fitting Euler spirals to points in [Raph Levien's PhD thesis](http://www.levien.com/phd/phd.html).2013-01-01

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Hermite interpolation is the beginning. This will give you a (cubic) curve that matches the two given points and tangents.

But it won't help much with your conditions A and C. To satisfy those, you need to ensure that your curve has "monotone curvature". If you look up that term, you will find lots of references.