I am interested in a class of (arc-length parametrized) curves $\gamma:\mathbb{R} \to \mathbb{R}^3$ with the following property:
- If the curve is written in cylindral coordinates $(r,\theta,z)$, it obeys the constraint $z'-r^2 \theta' \equiv c$ for some constant $c\in\mathbb{R}$. Here, $(\cdot)'$ represents the derivative of a coordinate wrt the arc-length parameter.
Obvious examples of curves obeying this constraint are helices where the (constant) ratio between $z'$ and $\theta'$ is exactly $r^2$, and thus $c=0$.
I'm interested in any article or book where the equation $z'-r^2\theta'\equiv c$, or even the special case $z'=r^2\theta'$, appears in reference to anything but a helix.
- Are there any other curves obeying this constraint that appear, e.g., as examples in differential-geometry textbooks?
- Has anyone seen this constraint appear anywhere else, e.g., in electromagnetism or some other area of physics?