// We know that LSegment[i-1] <= s < LSegment. 

public Vector3 InterpolatePositionByDistance(float s)
{
   //Assertions.Assert((s >= 0.0f) && (s <= Length));


   const int n = 20;  // step size

   float t = 0;
   float h = length / n;
   for (int i = 0; i < n; ++i)
   {
       // The divisions here might be a problem if the divisors are
       // nearly zero.
       float k1 = h / InterpolateDerivative(t).magnitude;
       float k2 = h / InterpolateDerivative(t + k1/2).magnitude;
       float k3 = h / InterpolateDerivative(t + k2/2).magnitude;
       float k4 = h / InterpolateDerivative(t + k3).magnitude;


       t += (k1 + 2 * (k2 + k3) + k4) / 6;
   }


   Debug.Log(t); // when s == length, and #segments > 1, t > 1 (for example t == 4.12334)


   return InterpolatePosition(t);
}