Here is the curve of a helix parametrized by its arc length $\alpha(s) = ( a\cos(\frac{s}{c}), a\sin(\frac{s}{c}), b(\frac{s}{c}) ), s \in \mathbb{R}$ such that a$^2$ + b$^2$ = c$^2$.
The curvature of the helix is k(s) = |$ \alpha ''$(s)| = $\frac{a}{c^2}$.
I am trying to figure out the meaning of curvature. What does it mean for the curvature to be equal to 1, greater than 1, and less than 1 (but of course greater than 0). i.e what happens to the curve in those three cases?
On
Let's rename $a=r$ and $b=h$ as usual. Consider a helix with curvature $\kappa$. It follows that $r/(r^2+h^2)=\kappa\iff h^2=r/\kappa-r^2$, that is $(r,h)$ lies on a circle centered at $(1/2\kappa,0)$ with radius $1/2\kappa$. Let's play with that!
What do helices with $\kappa=0.1$ look like? Well, $(r,h)$ could be $(2,4)$, $(5,5)$ or $(8,4)$. In general, for a given $\kappa$, $(r,h)$ could be $(1/(2\kappa),1/(2\kappa))$, that is, the maximum height of a helix for given $\kappa$ is $1/(2\kappa)$ whereas the maximum radius must be less than $1/\kappa$.
Or consider two helices with curvatures $\kappa_1\geq\kappa_2$ sharing the same radius $r$. Then $h_2^2-h_1^2=\left(\frac{1}{\kappa_2}-\frac{1}{\kappa_1}\right)r$.
Radius of curvature (the reciprocal of curvature) is much easier to understand. The radius of curvature at a point on a curve is equal to the radius of the so-called osculating circle at the point. This is the circle that most closely matches the curve at the point. See here and here for more details. So, to understand radius of curvature at a point, zoom in on that point and imagine the curve being approximated (locally) by a circle.
Radius of curvature can take any value between zero (a sharp corner) and infinity (a straight line).