Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js

Curvature k of a 1D function graph y = f(x)

Result

With f:RR κ(t)=|f

Derivation

We treat y = f(x) as a special case of a parametrized 2D curve \vec s(t) = [x(t), y(t)]:

\begin{aligned} x(t) &= t \\ y(t) &= f(t) \end{aligned}

Recall the formula of the curvature of a curve:

\kappa(t) = \frac{ \| s'(t) \times s''(t) \| }{ \| s'(t) \|^3 }

Unpack the terms for the 2D case:

\kappa = \frac{x'y'' - y'x''}{ \left( \sqrt{x'(t)^2 + y'(t)^2} \right)^3 }

Now we just plug in the parametric version of our function graph:

\begin{aligned} x'(t) &= 1 \\ x''(t) &= 0 \end{aligned}

\kappa(t) = \frac{|y''(t)|}{ \left ( \sqrt{1+y'(t)^2} \right)^3}

And voila.

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