I want a parametric equation of a sine wave at a small ramp angle wrapped around a cylindrical body (3D).
The parametric equation below gets me close to what I'm looking for, but not quite since the sine wave itself is not rotated (peaks still in line with the longitudinal direction of the cylindrical body).
$$x(t)=\sin(t)$$
$$y(t)=\cos(t)$$
$$z(t)=0.5t+\sin(4.5t)$$
In 2D, the following parametric equation will give me a rotated sine wave. How can I convert this to 3D around the cylinder following a helical pattern? The theta variable is the angle from the x-axis that the sine wave is rotated.
$$x(t)=t\cos(\theta)-\sin(\theta)\sin(t)$$
$$y(t)=t\sin(\theta)+\cos(\theta)\sin(t)$$
I appreciate any help you may be able to offer.
Thanks.
Best Answer
Try the following, where $t$ represents the main parameter, $\theta$ is the tilting angle as the sine wave wraps around the cylinder, $k$ is the frequency of the sine wave, and $a$ is the amplitude:
$$\begin{align*} x(t) &= \cos (t \cos \theta - a \sin \theta \sin (k t)) \\ y(t) &= \sin (t \cos \theta - a \sin \theta \sin (k t)), \\ z(t) &= t \sin \theta + a \cos \theta \sin (k t). \end{align*}$$
Fancy animated picture!