In order to calculate the required functions of a corkscrew, the projected arclength in the - plane, , will be used instead of the arclength along the corkscrew, . The conversion is made with the following, which can be inferred from figure 2.8.
The method of calculating a corkscrew will be to first create the circular and clothoidal regions in the - plane, as was done with a loop. The corkscrew will then be linearly stretched along the axis. The corkscrew will then be rotated downward radians. Figure 2.9 illustrates the final result of this process.
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 - plane, all that must be done is to negate the values of , thereby producing the mirror image.