The tangent to the helix at $P = (a \cos t, a \sin t, ct)$ is  

$T = (-a \sin t, a \cos t, c)$.  

The vector  

$H = (\cos t, \sin t, 0)$   

is horizontal, and perpendicular to $T$. The radius vector $R$ from $P$ to the centre $C$ of the circle is perpendicular to both of these, so it is proportional to  

$H \wedge T = (c \sin t, -c \cos t, a)$,

which has length $\sqrt{a^2 + c^2}$. So the centre of the circle is at $P + h H \wedge T$, with $h = r/\sqrt{a^2 + c^2}$.

A helical cycloid?
General Onimaru's Clockwork Soldier ability prevents players from moving backwards.  Does it also prevent retreating from a dashing strike?  If a player is forced to retreat past the Clockwork Soldier, does the Clockwork Soldier keep advancing and if so, is the player now prevented from moving past it in the opposite direction?
