Curvature

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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 $r$, the curvature at every point is $\frac{1}{r}$. Intuitively, this grows smaller as $r$ 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 $f(x)$, the curvature of the graph $y = f(x)$ of the function at the point $(x, f(x))$ is given by the formula $$\kappa(x) = \dfrac{f''(x)}{(f'(x)^2+1)^{3/2}}.$$

For a curve given in parametric form by the pair $(x(t), y(t))$, the curvature at a point is $$\kappa(t) = ?.$$ 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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