Back to the Corkscrew...

In order to calculate the required functions of a corkscrew, the projected arclength in the $x$-$z$ plane, $s_{proj}$, will be used instead of the arclength along the corkscrew, $s$. The conversion is made with the following, which can be inferred from figure 2.8.

$$s_{proj} = s \cos\alpha$$

The method of calculating a corkscrew will be to first create the circular and clothoidal regions in the $x$-$z$ plane, as was done with a loop. The corkscrew will then be linearly stretched along the $y$ axis. The corkscrew will then be rotated downward $\lambda$ radians. Figure 2.9 illustrates the final result of this process.

Figure 2.9: Corkscrew Construction.
These figures help to illustrate the construction of a corkscrew. The corkscrew is initially created in the $x$-$z$ plane. It is then linearly stretched along the $y$ axis. It is then tilted downward, to the $(x_{a},y_{a},z_{a})$ coordinates. The figure on the left shows the projection of the corkscrew onto the $(x_{a},z_{a})$ plane, which is perpendicular to the central axis. The figure on the right shows a side view of the corkscrew, clearly illustrating the downward tilt of the axis.

The equations of the corkscrew are derived assuming a counter-clockwise rotation. If a clockwise rotation is desired, the corkscrew will be a mirror image of the one calculated. Since the initial forward direction is in the $x$-$z$ plane, all that must be done is to negate the values of $y$, thereby producing the mirror image.