Is there a way to describe a helix not by its parametric form
$$ x=R\cos(t) ,\ y=R\sin(t) , \ z=ht , $$
but by a single equation like you can for a sphere with $ r^2 = x^2+y^2+z^2 $?
Also the same question for a 3-dimensional tube that follows a helix curve.
We can have asymmetric equation in terms of x, y and z:
To get such equation consider equation of a straight line in three dimension in form of symmetric:
$\frac{x-x_0}m=\frac{y-y_0}n=\frac{z-z_0}l$
To get parametric equation we can write:
$\frac{x-x_0}m=\frac{y-y_0}n=\frac{z-z_0}l=t$
$x=mt+x_0$, $y=nt+y_0$, $z=lt+z_0$
Clearly reverse procedure gives equation of line in terms of x, y and z. We apply this procedure to parametric equation of helix:
$x=R \cos(t)\Rightarrow t=\cos^{-1}\frac xR$
$y=R\sin (t)\Rightarrow t=\sin^{-1}\frac yR$
$z=ht \Rightarrow t=\frac z h$
Therefore symmetric equation for helix can be:
$\cos^{-1}\frac xR=\sin^{-1}\frac yR=\frac zh$