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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?

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  • $\begingroup$ Smaller the curvature, closer it is to being a straight line. The curvature of any curve is dependent on the surface of which it is a part of.Can you elaborate on your question??I dont see what is special about 1 ?? $\endgroup$ – Vishesh Jan 23 '14 at 4:58
  • $\begingroup$ what is special about constant curvature? what curves have constant curvature? circle and helix? $\endgroup$ – sarah Jan 23 '14 at 5:02
  • $\begingroup$ Yes in Euclidean 3-space that is all. There is a neat result that proves this. But in curved spaces (or Manifolds as they are called) there are others. Well a constantly curving object has some nice properties, that is a crude answer, but this is something better explored rather than answered. I can elaborate if you still have doubts. $\endgroup$ – Vishesh Jan 23 '14 at 5:08
  • $\begingroup$ so there is nothing special about the value of constant curvature except when it is close to 0? what if the curvature is really large but still constant? for example how does the curve look like if curvature is 0.000000001 versus curvature 10000000000? $\endgroup$ – sarah Jan 23 '14 at 5:18
  • $\begingroup$ If you really want to get good illustrations, I would suggest Mathematica. However from a theoretical perspective, constant curvature means helix or circle or some part of one of these in flat space. Well the first curve looks almost like a straight line while the second if it is a constant curvature curve will be part of a circle or helix of huge radius, that is all. $\endgroup$ – Vishesh Jan 23 '14 at 5:23
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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).

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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 $h_2^2-h_1^2=\left(\frac{1}{\kappa_2}-\frac{1}{\kappa_1}\right)r$.

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To get a comparative feel, compare the curvature of the cylinder 1/a with the normal curvature of helix a/c^2. If a = c, the helix has same amount of normal curvature at 45 degrees helix angle obtained by a cutting plane along tangent in normal direction.

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