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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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You've looked up Hermite interpolation? – J. M. Feb 16 '12 at 15:37
@J.M. No. Although I knew the name , cspline, I had no idea of that it was the one I looked for . Thank you very much. – seven_swodniw Feb 16 '12 at 16:20
The special case of curvature changing linearly with length along the curve is called an Euler spiral. You might find some information about fitting Euler spirals to points in Raph Levien's PhD thesis. – Rahul Jan 1 '13 at 9:03

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.

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