What's is the equation of sinusoidal function on cylinder perimeter

Question

How can I determine the equation of a sinusoidal function that instead of proceeding on a horizontal line, proceeds along the perimeter of a cylinder while its sinus also lies on the surface area of the cylinder, like radially squeezing the cylinder?

Answer 1

Just use cylindrical coordinates $$ z = h\sin \left( {\lambda \theta + \alpha } \right) $$

where:
- $h$ is the amplitude of the sinusoid; - $r$ is the radius of the circular cylinder around the $z$ axis;
- $\lambda$ is the angular speed of the sinusoid (n. of repetitions along one turn around the cylinder);
- $\alpha$ is the phase (in rad) of the sinusoid;
- $\theta$ is the parameter, which is the angle (in rad) of the generatrix of the cylinder.

You can then convert to a parametric equation in Euclidean coordinates as $$ \left\{ \matrix{ x = r\cos \theta \hfill \cr y = r\sin \theta \hfill \cr z = h\sin \left( {\lambda \theta + \alpha } \right) \hfill \cr} \right. $$

example: