A non-rigorous explanation of curvature of curves in 3-space

In a second year calculus class there is usually the definition of curvature as $$\kappa=\left|\frac{d\bf{T}}{ds}\right|$$where $\bf{T}\rm(t)=\frac{r^\prime(t)}{|r^\prime(t)|}$ is the unit tangent vector and $s$ is the arc length. The curvature is described as a measure of how quickly the curve changes direction at a point. From the above formula, it is derived that $$\kappa(t)=\frac{\left|\bf{T}\rm^\prime(t)\right|}{\left|r^\prime(t)\right|}$$I am failing to see how either of these two formulas is measuring how quickly the curve changes direction. Taking the derivative with respect to arc-length just completely blows away my intuition. Where-as the ratio of the rate of change of tangent vectors by rate of change of the curve seems more plausible. However, it still isn't clear. How can you even calculate this desired quantity at a specific point, for to know how quickly the curve changes at a point you would need to know what the curve is doing before and/or after that point.

Basically, I am hoping that someone can explain to me, non-rigorously, the geometric intuition of curvature of a curve in $3$-space. From this description, how do we arrive at one of the two formulas?

• Could you please do the calculation for $r= (5 \cos t, 5\sin t,0)$ – Will Jagy Sep 23 '14 at 21:48
• Only because you asked nicely. $T(t)=(-\sin t,\cos t,0)$ so that $T^\prime(t)=(-\cos t,-\sin t,0)$. Then $|T^\prime(t)|=1$ while $|r^\prime(t)|=\sqrt{25}=5$. So $\kappa=\frac{1}{5}$. – user162520 Sep 23 '14 at 21:54
• which is the reciprocal of the radius of the best-fitting circle. For plane curves other than circles, that still holds. For space curves, still true with the proviso that we need also decide in what plane the best circle should lie, and that is the plane through the point in question spanned by $T,N$ – Will Jagy Sep 23 '14 at 21:57
• So why do we define the curvature of a circle to be the reciprocal the radius? – user162520 Sep 23 '14 at 21:59
• I wasn't there when they decided that. I probably would have voted in favor, though. – Will Jagy Sep 23 '14 at 22:01