Editing Curvature (section)

=== Examples ===
It can be useful to verify on simple examples that the different formulas given in the preceding sections give the same result.

====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>

====Parabola====
Consider the [[parabola]] {{math|''y'' {{=}} ''ax''<sup>2</sup> + ''bx'' + ''c''}}.

It is the graph of a function, with derivative {{math|2''ax'' + ''b''}}, and second derivative {{math|2''a''}}. So, the signed curvature is 
:<math>k(x)=\frac{2a}{ \bigl(1+\left(2ax+b\right)^2\bigr)\vphantom{)}^{3/2}}.</math>
It has the sign of {{mvar|a}} for all values of {{mvar|x}}. This means that, if {{math|''a'' > 0}}, the concavity is upward directed everywhere; if {{math|''a'' < 0}}, the concavity is downward directed; for {{math|1=''a'' = 0}}, the curvature is zero everywhere, confirming that the parabola degenerates into a line in this case.

The (unsigned) curvature is maximal for {{math|1=''x'' = –{{sfrac|''b''|2''a''}}}}, that is at the [[stationary point]] (zero derivative) of the function, which is the [[vertex (curve)|vertex]] of the parabola.

Consider the parametrization {{math|'''γ'''(''t'') {{=}} (''t'', ''at''<sup>2</sup> + ''bt'' + ''c'') {{=}} (''x'', ''y'')}}. The first derivative of {{mvar|x}} is {{math|1}}, and the second derivative is zero. Substituting into the formula for general parametrizations gives exactly the same result as above, with {{mvar|x}} replaced by {{mvar|t}}. If we use primes for derivatives with respect to the parameter {{mvar|t}}.

The same parabola can also be defined by the implicit equation {{math|1= ''F''(''x'', ''y'') = 0}} with {{math|''F''(''x'', ''y'') {{=}} ''ax''<sup>2</sup> + ''bx'' + ''c'' – ''y''}}. As {{math|1=''F{{sub|y}}'' = –1}}, and {{math|1=''F{{sub|yy}}'' = ''F{{sub|xy}}'' = 0}}, one obtains exactly the same value for the (unsigned) curvature. However, the signed curvature is meaningless here, as {{math|1=–''F''(''x'', ''y'') = 0}} is a valid implicit equation for the same parabola, which gives the opposite sign for the curvature.