Axis Calculations

The central axis length of the corkscrew is similar to the shift angle of the loop or the vertical angle of a curve. It is the direction the corkscrew is ``linearly stretched'' along to create the three-dimensional element. The distance along the central axis varies linearly with the arclength along the corkscrew.

Let $s_{\beta}$ represent the projected arclength in the $x$-$z$ plane of the corkscrew from $\phi = \pi/2$ to $\phi = \pi$. Let $c$ represent the length along the axis over the same range of $\phi$. Refer to figure 2.7. Since the axis length varies linearly as the arclength along the corkscrew, and therefore varies linearly as the projected arclength along the corkscrew,^2.7 the following can be shown.

$$\begin{eqnarray*} \frac{c}{s_{\beta}} & = & \frac{l_{a}}{l_{proj}} \\ l_{a} & = & \left( \frac{c}{s_{\beta}} \right) l_{proj} \end{eqnarray*}$$

The range of $s_{\beta}$ is entirely within the circular region of the corkscrew. Its projected arclength in the $x$-$z$ plane can therefore be determined from the following.

$$s_{\beta} = r \left( \frac{\pi}{2} \right)$$

From the figure a relationship can be found for $c$ in terms of $\beta$.

$$c = r \tan \beta$$

Making substitutions from above determines the total axis length.

$$\begin{eqnarray*} l_{a} & = & \frac{r\tan\beta}{r(\pi/2)} l_{proj} \\ & = & \frac{2 l_{proj} \tan\beta }{\pi} \end{eqnarray*}$$

Figure 2.8: Corkscrew Shift Angle.
This figure illustrates the relationship between the total length of the corkscrew, the length of the projection in the $x$-$z$ plane, and the axis length. Note that $\alpha$ is a constant angle throughout the corkscrew, as explained in section 2.1.

The shift angle of a corkscrew, $\alpha$, is analogous to that of a loop. Figure 2.8 depicts the arclength of the corkscrew as the hypotenuse of the triangle, while the lower leg represents the projected length in the $x$-$z$ plane. The length of the axis is the length of the right leg.

$$\alpha = \arctan \left( \frac{l_{a}}{l_{proj}} \right)$$

The arclength of the corkscrew is given by the Pythagorean Theorem.

$$l = \sqrt{l_{proj}^{2} + l_{a}^{2}}$$

^2.7 Refer to figure 2.8. Since the angle $\alpha$ is constant, as discussed in section 2.1, this is a true statement.