The end of the first clothoid occurs when . The position at is found, and the center of the circular region is determined from it. This point, , can be seen in figure 2.5.
Derivatives with respect to arclength will need to be taken to determine the remaining functions. Since the position vector is expressed in terms of , the chain rule is applied.
The forward vector is calculated as the derivative of the position vector with respect to
The radial vector is calculated as the derivative of the forward vector with respect to arclength.