Editing Curvature (section)

===Frenet–Serret formulas for plane curves===
[[File:FrenetTN.svg|thumb|right|350px|The vectors {{math|'''T'''}} and {{math|'''N'''}} at two points on a plane curve, a translated version of the second frame (dotted), and {{math|''δ'''''T'''}} the change in {{math|'''T'''}}. Here {{mvar|δs}} is the distance between the points. In the limit {{math|{{sfrac|''d'''''T'''|''ds''}}}} will be in the direction {{math|'''N'''}}. The curvature describes the rate of rotation of the frame.]]
The expression of the curvature [[#In terms of arc-length parametrization|In terms of arc-length parametrization]] is essentially the [[Frenet–Serret formulas|first Frenet–Serret formula]]
:<math>\mathbf T'(s) = \kappa(s) \mathbf N(s),</math>
where the primes refer to the derivatives with respect to the arc length {{mvar|s}}, and {{math|'''N'''(''s'')}} is the normal unit vector in the direction of {{math|'''T'''′(s)}}.

As planar curves have zero [[torsion of curves|torsion]], the second Frenet–Serret formula provides the relation
:<math>\begin{align}
\frac {d\mathbf{N}}{ds} &= -\kappa\mathbf{T},\\
            &= -\kappa\frac{d\boldsymbol{\gamma}}{ds}.
\end{align}</math>

For a general parametrization by a parameter {{mvar|t}}, one needs expressions involving derivatives with respect to {{mvar|t}}. As these are obtained by multiplying by {{sfrac|''ds''|''dt''}} the derivatives with respect to {{mvar|s}}, one has, for any proper parametrization
:<math>
\mathbf{N}'(t) = -\kappa(t)\boldsymbol{\gamma}'(t).
</math>