Editing Curvature (section)

====Circle====
A common parametrization of a [[circle]] of radius {{mvar|r}} is {{math|1='''γ'''(''t'') = (''r'' cos ''t'', ''r'' sin ''t'')}}. The formula for the curvature gives
:<math>k(t)= \frac{r^2\sin^2 t + r^2\cos^2 t}{\bigl(r^2\cos^2 t+r^2\sin^2 t\bigr)\vphantom{'}^{3/2}} = \frac 1r.</math>

It follows, as expected, that the radius of curvature is the radius of the circle, and that the center of curvature is the center of the circle.

The circle is a rare case where the arc-length parametrization is easy to compute, as it is
:<math>\boldsymbol\gamma(s)= \left(r\cos \frac sr,\, r\sin \frac sr\right).</math>
It is an arc-length parametrization, since the norm of
:<math>\boldsymbol\gamma'(s) = \left(-\sin \frac sr,\, \cos \frac sr\right)</math>
is equal to one. This parametrization gives the same value for the curvature, as it amounts to division by {{math|''r''<sup>3</sup>}} in both the numerator and the denominator in the preceding formula.

The same circle can also be defined by the implicit equation {{math|1= ''F''(''x'', ''y'') = 0}} with {{math|1=''F''(''x'', ''y'') = ''x''{{sup|2}} + ''y''{{sup|2}} – ''r''{{sup|2}}}}. Then, the formula for the curvature in this case  gives
:<math>\begin{align}
\kappa &= \frac{\left|F_y^2F_{xx}-2F_xF_yF_{xy}+F_x^2F_{yy}\right|}{\bigl(F_x^2+F_y^2\bigr)\vphantom{'}^{3/2}}\\
&=\frac{8y^2 + 8x^2}{\bigl(4x^2+4y^2\bigr)\vphantom{'}^{3/2}}\\
&=\frac {8r^2}{\bigl(4r^2\bigr)\vphantom{'}^{3/2}} =\frac1r.\end{align}</math>