Helixes

A helix looks a lot like a spring. There's an axis, along which the spring is stretched out. Let's let this be the z axis. The cross sections of the helix look like circles almost, except they they start at one value of z and wind up at a different value after coming ``full circle.''

One way to generate this from a parameterization is to look at the path as being in two parts. In the xy part, we want a circle (or an ellipse):

$$\vec{r}_{xy}(t) = a \cos t \hat{i} + a \sin tt \hat{j}.$$ (23)

In the z direction, we basically want a line:

$$\vec{r}_z(t) = t \hat{k}.$$ (24)

Thus, the total motion can be written as the sum of these two pieces of the motion:

$$\vec{r}(t) = a \cos t \hat{i} + a \sin t \hat{j} + t \hat{k}.$$ (25)