Is this spiral known?

Question

Parametrized as

$$ \sec \theta \,( p \cos(\theta+ \alpha), \,p \sin(\theta+ \alpha) , c\alpha), $$

the spiral is plotted $ (-\pi/4<\theta< \pi/4;\,\,0< \alpha < 3 \pi) $ for $ p= 1$ and $ c=0.125 $ on Mathematica.

Answer 1

Not aware that I've seen this specific surface elsewhere, but (as you may know) it's obtained, for $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$, from one branch of a "steepening" the right hyperbola lying in a plane at distance $p$ from the $z$-axis by carrying it through space via a helical motion, since $$ \sec\theta \bigl(p \cos(\theta + \alpha), p \sin(\theta + \alpha), c\alpha\bigr) = p(\cos\alpha, \sin\alpha, 0) + (-\tan\theta \sin\alpha, \tan\theta \cos\alpha, c\alpha\sec\theta). $$