Curvature
Curvature is a a number associated with every point on each smooth curve that describes "how curvy" the curve is at that point. In particular, the "least curvy" curve is a line, and fittingly lines have zero curvature. For a circle of radius 
, the curvature at every point is 
. Intuitively, this grows smaller as grows larger because one must turn much more sharply to follow the path of a circle of small radius than to follow the path of a circle with large radius.
Given a twice-differentiable function 
, the curvature of the graph 
of the function at the point 
is given by the formula
![\[\kappa(x) = \dfrac{f''(x)}{(f'(x)^2+1)^{3/2}}.\]](http://latex.artofproblemsolving.com/d/0/b/d0b863969c855002143f753e178e5cab7e8a54ba.png)
For a curve given in parametric form by the pair 
, the curvature at a point is
![\[\kappa(t) = ?.\]](http://latex.artofproblemsolving.com/e/a/8/ea800c6bf2dc26c012c873e3d7086306adde13b9.png)
This expression is invariant under positive-velocity reparametrizations, that is the curvature is a property of the curve and not the way in which you traverse it.
Curvature of surfaces
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