Editing Curvature (section)

===In terms of arc-length parametrization===
Every [[differentiable curve]] can be [[parametric representation|parametrized]] with respect to [[arc-length parametrization|arc length]].<ref>{{citation |type=Website
 | url = https://sites.google.com/site/johnkennedyshome/home/class-downloads
 | last1 = Kennedy
 | first1 = John
 | title = The Arc Length Parametrization of a Curve
 | year = 2011
 | access-date = 2013-12-10
 | archive-date = 2015-09-28
 | archive-url = https://web.archive.org/web/20150928030020/https://sites.google.com/site/johnkennedyshome/home/class-downloads
 | url-status = dead
 }}</ref> In the case of a plane curve, this means the existence of a parametrization {{math|'''γ'''(''s'') {{=}} (''x''(''s''), ''y''(''s''))}}, where {{mvar|x}} and {{mvar|y}} are real-valued differentiable functions whose derivatives satisfy
:<math>\|\boldsymbol{\gamma}'\| = \sqrt{x'(s)^2+y'(s)^2} = 1.</math>
This means that the tangent vector
:<math>\mathbf T(s)=\bigl(x'(s),y'(s)\bigr)</math>
has a length equal to one and is thus a [[unit tangent vector]].

If the curve is twice differentiable, that is, if the second derivatives of {{mvar|x}} and {{mvar|y}} exist, then the derivative of {{math|'''T'''(''s'')}} exists. This vector is normal to the curve, its length is the curvature {{math|''κ''(''s'')}}, and it is oriented toward the center of curvature. That is,
:<math>\begin{align}
\mathbf{T}(s) &= \boldsymbol{\gamma}'(s), \\[8mu]
\|\mathbf{T}(s)\|^2 &= 1 \ \text{(constant)} \implies \mathbf{T}'(s)\cdot \mathbf{T}(s) = 0, \\[5mu]
\kappa(s) &= \|\mathbf{T}'(s)\| = \|\boldsymbol{\gamma}''(s)\| = \sqrt{x''(s)^2+y''(s)^2}
\end{align}</math>

Moreover, because the radius of curvature is (assuming ''𝜿''(''s'') ≠ 0)
:<math>R(s)=\frac{1}{\kappa(s)},</math>
and the center of curvature is on the normal to the curve, the center of curvature is the point 
:<math> \mathbf{C}(s)= \boldsymbol{\gamma}(s) + \frac 1{\kappa(s)^2}\mathbf{T}'(s).</math> 
(In case the curvature is zero, the center of curvature is not located anywhere on the plane '''''R'''''<sup>2</sup> and is often said to be located "at infinity".)

{{anchor|signed curvature}}
If {{math|'''N'''(''s'')}} is the [[unit normal vector]] obtained from {{math|'''T'''(''s'')}} by a counterclockwise rotation of {{sfrac|{{pi}}|2}}, then 
:<math>\mathbf{T}'(s)=k(s)\mathbf{N}(s),</math>
with {{math|1= ''k''(''s'') = ± ''κ''(''s'')}}. The real number {{math|''k''(''s'')}} is called the '''oriented curvature''' or '''signed curvature'''. It depends on both the orientation of the plane (definition of counterclockwise), and the orientation of the curve provided by the parametrization. In fact, the [[change of variable]] {{math|''s'' → –''s''}}   provides another arc-length parametrization, and changes the sign of {{math|''k''(''s'')}}.

With the above, the center of curvature can be expressed as: 
:<math>\mathbf{C}(s)= \boldsymbol{\gamma}(s) + 
R(s)\mathbf{N}(s).</math>