Curvature k of a 1D function graph y = f(x)
How to compute the curvature of a function graph with one variable. - 04/2024 - #Jumble
Result
With $f:\mathbb R \rightarrow \mathbb R$ $$ \kappa(t) = \frac{|f''(t)|}{ \left ( \sqrt{1+f'(t)^2} \right)^3} $$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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