All Regions

As with a loop, a corkscrew is initially created with a non-zero initial azimuth angle. As explained in section 2.1, the azimuth angle must be zero at the start of every element. This is required in order to guarantee slope continuity when the elements are joined. Since the corkscrew has a non-zero initial azimuth angle, it must be rotated about the $z$ axis. This will result in both the initial and final azimuth angles being zero.

$$\theta_{f} = 0$$ (2.114)

The value of $\eta$ is the initial angle in the $x$-$y$ plane. It is found by evaluating the forward vector of clothoid 1 at $s=0$, where $\phi = 0$. This results in the following.

$$\begin{eqnarray*} f_{x} & = & \cos\alpha \\ f_{y} & = & \cos\phi \sin\alpha \end{eqnarray*}$$

The value of $\eta$ can now be determined.

$$\begin{eqnarray*} \eta & = & \arctan \left( \frac{f_{y}}{f_{x}} \right) \\ & = & \arctan \left( \cos\phi \tan\alpha \right) \end{eqnarray*}$$

The corkscrew can now be rotated $-\eta$ radians about the $z$ axis to force it to begin in the $x$-$z$ plane. The rotation is performed as follows, where the subscript $r$ stands for rotated, and the subscript $nr$ stands for non-rotated.

$$\begin{eqnarray*} x_{r} & = & x_{nr} \cos(-\eta) - y_{nr} \sin(-\eta) \\ & = ... ...& = & -x_{nr} \sin\eta + y_{nr} \cos\eta \\ z_{r} & = & z_{nr} \end{eqnarray*}$$

The rotation must be performed on the position vectors, forward vectors, and radial vectors. A more complete description of rotation transformations is given in appendix B.

The calculation of the initial vertical angle requires the use of the forward vector at the beginning of the corkscrew. The following is obtained by taking derivatives of the position vector after all rotations.

$$\begin{eqnarray*} f_{x} & = & \cos\phi \cos\alpha \cos\eta + \sin\alpha \cos\la... ...} & = & \sin\alpha \sin\lambda + \sin\phi \cos\alpha \cos\lambda \end{eqnarray*}$$

The value of $\phi_{0}$ is the angle between the forward vector and the $x$ axis at the beginning of the corkscrew, when $\phi = 0$.

$$\phi_{0} = \arctan \left( \frac{f_{z}}{f_{x}} \right)$$

$$= \arctan \left( \frac{ \sin\lambda \sin\alpha }{ \cos\alpha \cos\eta + \cos\lambda \sin\alpha \sin\eta } \right)$$ (2.115)

The result is the same for the final vertical angle, $\phi_{f}$.