How to mathematically formulate the surface of a spring?

Question

I would like to mathematically map the surface of a cylinder constructed like a coil pot (or compressed spring), where the surface area and height of the pot is a function of the length of the coil, and the length of the coil is a function of time. The cylinder is infinitely long. Is there an existing way to do this?

Answer 1

Axis-symmetric surface of torus can be shear deformed by adding a torsion term to $z-$ coordinate of a circular symmetric torus as $c \cdot\theta$.

$$ (x,y,z)= [ (a+ b \cos \phi ) \cos\theta,\,(a+ b \cos\phi )\sin\theta ,\, b \sin \phi + c \cdot \theta\,] $$

$a$ is spring radius, $b$ is coil radius, $ c $ is torsional radius of curvature.

The above is written with respect to $\theta$. To convert to arc length$ s$ of coil middle line, use:

$$ \dfrac{d \theta}{ds} = \sin \alpha/ a, \tan \alpha = a/c $$