Editing Curvature (section)

=== In terms of a general parametrization ===
Let {{math|'''γ'''(''t'') {{=}} (''x''(''t''), ''y''(''t''))}} be a proper [[parametric representation]] of a twice differentiable plane curve. Here ''proper'' means that on the [[domain of a function|domain]] of definition of the parametrization, the derivative {{math|{{sfrac|''d'''''γ'''|''dt''}}}}
is defined, differentiable and nowhere equal to the zero vector.

With such a parametrization, the signed curvature is 
:<math>k = \frac{x'y''-y'x''}{\bigl({x'}^2+{y'}^2\bigr)\vphantom{'}^{3/2}},</math>
where primes refer to derivatives with respect to {{mvar|t}}. The curvature {{mvar|''κ''}} is thus
:<math>\kappa = \frac{\left|x'y''-y'x''\right|}{\bigl({x'}^2+{y'}^2\bigr)\vphantom{'}^{3/2}}.</math>

These can be expressed in a coordinate-free way as
:<math>
k = \frac{\det\left(\boldsymbol{\gamma}',\boldsymbol{\gamma}''\right)}{\|\boldsymbol{\gamma}'\|^3},\qquad
\kappa = \frac{\left|\det\left(\boldsymbol{\gamma}',\boldsymbol{\gamma}''\right)\right|}{\|\boldsymbol{\gamma}'\|^3}.
</math>

These formulas can be derived from the special case of arc-length parametrization in the following way. The above condition on the parametrisation imply that the arc length {{mvar|s}} is a differentiable [[monotonic function]] of the parameter {{mvar|t}}, and conversely that {{mvar|t}} is a monotonic function of {{mvar|s}}. Moreover, by changing, if needed, {{mvar|s}} to {{math|–''s''}}, one may suppose that these functions are increasing and have a positive derivative. Using notation of the preceding section and the [[chain rule]], one has 
:<math>\frac{d\boldsymbol{\gamma}}{dt}= \frac{ds}{dt}\mathbf T,</math>
and thus, by taking the norm of both sides 
:<math> \frac{dt}{ds}= \frac 1{\|\boldsymbol{\gamma}'\|},</math>
where the prime denotes differentiation with respect to {{mvar|t}}.

The curvature is the norm of the derivative of {{math|'''T'''}} with respect to {{mvar|s}}. By using the above formula and the chain rule this derivative and its norm can be expressed in terms of {{math|'''γ'''′}} and {{math|'''γ'''″}} only, with the arc-length parameter {{mvar|s}} completely eliminated, giving the above formulas for the curvature.