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.
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.