Exact definition of a circular helix

I know that a cylindrical helix is a curve $\alpha: I \to \Bbb R^3$, such that exists a constant vector $\bf u$ which makes a constant angle with the tangent vector $\bf T$ to the curve, at every point, that is, $\langle {\bf T}(s), {\bf u}\rangle = \cos \theta$, for all $s \in I$ (we can suppose that $\alpha$ is parametrized by arc-length).

Then, there this a caracterization of cylindrical helices: a curve $\alpha$ with non-zero curvature is a cylindrical helix if and only if $\tau/\kappa$ is constant. Notice that $\kappa$ and $\tau$ need not be constant, only their ratio, though.

Now, it is stated:

A curve $\alpha$ is a circular helix if and only if both $\kappa$ and $\tau$ are constants.

How am I supposed to prove this? Every book seems to think that it is so obvious what a circular helix is, but no one gives a straight definition for it. I am at a loss about what to do. I might as well define a circular helix being a curve with this property. If someone could tell me an exact definition for it, or give me a reference, I am thankful.

Note: I found this question, but I don't think it was intended for the proof of this affirmation to become as complicated as in the answer there.

A cylindrical helix is a curve on a generalized cylinder (take any space curve and form a cylinder by taking parallel lines through each point of the space curve) that makes a constant angle with the rulings (the parallel lines). A circular helix is such a curve on a right circular cylinder. Any such is congruent to a standard helix: $$\alpha(t)=(a\cos t, a \sin t, bt)$$ for some $a>0$, $b\ne 0$.