Editing Curvature (section)

==Plane curves==
Intuitively, the curvature describes for any part of a curve how much the curve direction changes over a small distance travelled (e.g. angle per length in [[Radians per metre|{{math|rad/m}}]]), so it is a measure of the [[instantaneous rate of change]] of ''direction'' of a point that moves on the curve: the larger the curvature, the larger this rate of change. In other words, the curvature measures how fast the unit tangent vector to the curve at point {{mvar|P}} rotates<ref>{{citation |last=Pressley |first=Andrew |url=https://books.google.com/books?id=UXPyquQaO6EC |title=Elementary Differential Geometry |date=2001 |publisher=Springer |isbn=978-1-85233-152-8 |place=London |page=29 }}</ref> when point {{mvar|P}} moves at unit speed along the curve. In fact, it can be proved that this instantaneous rate of change of direction is exactly the curvature. 

More precisely, suppose that the point is moving on the curve at a constant speed of one unit, that is, the position of the point {{math|''P''(''s'')}} is a function of the parameter {{mvar|s}}, which may be thought as the time or as the [[arc length]] from a given origin. Let {{math|'''T'''(''s'')}} be a [[unit tangent vector]] of the curve at {{math|''P''(''s'')}}, which is also the [[derivative]] of {{math|''P''(''s'')}} with respect to {{mvar|s}}. Then, the derivative of {{math|'''T'''(''s'')}} with respect to {{mvar|s}} is a vector that is normal to the curve and whose length is the curvature.

To be meaningful, the definition of the curvature and its different characterizations require that the curve is [[continuously differentiable]] near {{mvar|P}}, for having a tangent that varies continuously; it requires also that the curve is twice differentiable at {{mvar|P}}, for insuring the existence of the involved limits, and of the derivative of {{math|'''T'''(''s'')}}.

The characterization of the curvature in terms of the derivative of the unit tangent vector is probably less intuitive than the definition in terms of the osculating circle, but formulas for computing the curvature are easier to deduce. Therefore, and also because of its use in [[kinematics]], this characterization is often given as a definition of the curvature.

===Osculating circle===
[[File:Osculating.svg|alt=|right|250x250px]]
Historically, the curvature of a differentiable curve was defined through the [[osculating circle]], which is the circle that best approximates the curve at a point. More precisely, given a point {{mvar|P}} on a curve, every other point {{mvar|Q}} of the curve defines a circle (or sometimes a line) passing through {{mvar|Q}} and [[tangent (geometry)|tangent]] to the curve at {{mvar|P}}. The osculating circle is the [[limit (mathematics)|limit]], if it exists, of this circle when {{mvar|Q}} tends to {{mvar|P}}. Then the ''center'' and the ''radius of curvature'' of the curve at {{mvar|P}} are the center and the radius of the osculating circle. The curvature is the [[multiplicative inverse|reciprocal]] of radius of curvature. That is, the curvature is 
: <math> \kappa = \frac{1}{R},</math>
where {{mvar|R}} is the radius of curvature<ref>{{harvnb|Kline|1998|page=458}}</ref> (the whole circle has this curvature, it can be read as turn {{math|2π}} over the length {{math|2π}}{{mvar|R}}).

This definition is difficult to manipulate and to express in formulas. Therefore, other equivalent definitions have been introduced.

===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>

=== 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.

===Graph of a function===
The [[graph of a function]] {{math|''y'' {{=}} ''f''(''x'')}}, is a special case of a parametrized curve, of the form
:<math>\begin{align}
x&=t\\
y&=f(t).
\end{align}</math>
As the first and second derivatives of {{mvar|x}} are 1 and 0, previous formulas simplify to 
:<math>\kappa = \frac{\left|y''\right|}{\bigl(1+{y'}^2\bigr)\vphantom{'}^{3/2}},</math>
for the curvature, and to
:<math>k = \frac{y''}{\bigl(1+{y'}^2\bigr)\vphantom{'}^{3/2}},</math>
for the signed curvature.

In the general case of a curve, the sign of the signed curvature is somewhat arbitrary, as it depends on the orientation of the curve. In the case of the graph of a function, there is a natural orientation by increasing values of {{mvar|x}}. This makes significant the sign of the signed curvature.

The sign of the signed curvature is the same as the sign of the second derivative of {{mvar|f}}. If it is positive then the graph has an upward concavity, and, if it is negative the graph has a downward concavity. If it is zero, then  one has an [[inflection point]] or an [[undulation point]].

When the [[slope]] of the graph (that is the derivative of the function) is small, the signed curvature is well approximated by the second derivative. More precisely, using [[big O notation]], one has 
:<math>k(x)=y'' \Bigl(1 + O\bigl({\textstyle y'}^2\bigr) \Bigr).</math>
 
It is common in [[physics]] and [[engineering]] to approximate the curvature with the second derivative, for example, in [[beam theory]] or for deriving the [[wave equation]] of a string under tension, and other applications where small slopes are involved. This often allows systems that are otherwise [[Nonlinear system|nonlinear]] to be treated approximately as linear.

===Polar coordinates===

If a curve is defined in [[polar coordinates]] by the radius expressed as a function of the polar angle, that is {{mvar|r}} is a function of {{mvar|θ}}, then its curvature is
:<math>\kappa(\theta) = \frac{\left|r^2 + 2{r'}^2 - r\, r''\right|}{\bigl(r^2+{r'}^2 \bigr)\vphantom{'}^{3/2}}</math>
where the prime refers to differentiation with respect to {{mvar|θ}}.

This results from the formula for general parametrizations, by considering the parametrization 
:<math>\begin{align}
x&=r\cos \theta\\
y&=r\sin \theta
\end{align}</math>

=== Implicit curve ===

For a curve defined by an [[implicit equation]] {{math|''F''(''x'', ''y'') {{=}} 0}} with [[partial derivatives]] denoted  {{mvar|F<sub>x</sub>}} , {{mvar|F<sub>y</sub>}} , {{mvar|F<sub>xx</sub>}} , {{mvar|F<sub>xy</sub>}} , {{mvar|F<sub>yy</sub>}} ,
the curvature is given by<ref>{{citation |last=Goldman |first=Ron |year=2005 |title=Curvature formulas for implicit curves and surfaces |journal=Computer Aided Geometric Design |volume=22 |issue=7 |pages=632–658 |citeseerx=10.1.1.413.3008 |doi=10.1016/j.cagd.2005.06.005}}</ref>
:<math>\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}}.</math>

The signed curvature is not defined, as it depends on an orientation of the curve that is not provided by the implicit equation. Note that changing {{mvar|F}} into {{math|–''F''}} would not change the curve defined by {{math|''F''(''x'', ''y'') {{=}} 0}}, but it would change the sign of the numerator if the absolute value were omitted in the preceding formula.

A point of the curve where {{math|1=''F<sub>x</sub>'' = ''F<sub>y</sub>'' = 0}} is a [[singular point of a curve|singular point]], which means that the curve is not differentiable at this point, and thus that the curvature is not defined (most often, the point is either a crossing point or a [[cusp (singularity)|cusp]]).

The above formula for the curvature can be derived from the expression of the curvature of the graph of a function by using the [[implicit function theorem]] and the fact that, on such a curve, one has 
:<math>\frac {dy}{dx}=-\frac{F_x}{F_y}.</math>

=== 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.

===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>

===Curvature comb===
[[File:Curvature comb.png|alt=Curvature comb|thumb|248x248px|Curvature comb]]
A ''curvature comb''<ref>{{citation |last=Farin |first=Gerald |date=November 2016 |title=Curvature combs and curvature plots |journal=Computer-Aided Design |volume=80 |pages=6–8 |doi=10.1016/j.cad.2016.08.003}}</ref> can be used to represent graphically the curvature of every point on a curve. If <math>t \mapsto x(t)</math> is a parametrised curve its comb is defined as the parametrized curve
:<math> t \mapsto x(t) + d\kappa(t)n(t)</math>
where <math>\kappa, n</math> are the curvature and normal vector and <math>d</math> is a scaling factor (to be chosen as to enhance the graphical representation).