Editing Curvature (section)

==Space curves==
[[File:Torus-Knot uebereinander animated.gif|thumb|upright|Animation of the curvature and the acceleration vector {{math|'''T'''′(''s'')}}]]
As in the case of curves in two dimensions, the curvature of a regular [[space curve]] {{mvar|C}} in three dimensions (and higher) is the magnitude of the acceleration of a particle moving with unit speed along a curve. Thus if {{math|'''γ'''(''s'')}} is the arc-length parametrization of {{mvar|C}} then the unit tangent vector {{math|'''T'''(''s'')}} is given by
:<math>\mathbf{T}(s) = \boldsymbol{\gamma}'(s)</math>
and the curvature is the magnitude of the acceleration:
:<math>\kappa(s) = \|\mathbf{T}'(s)\| = \|\boldsymbol{\gamma}''(s)\|.</math>
The direction of the acceleration is the unit normal vector {{math|'''N'''(''s'')}}, which is defined by
:<math>\mathbf{N}(s) = \frac{\mathbf{T}'(s)}{\|\mathbf{T}'(s)\|}.</math>

The plane containing the two vectors {{math|'''T'''(''s'')}} and {{math|'''N'''(''s'')}} is the [[osculating plane]] to the curve at {{math|'''γ'''(''s'')}}. The curvature has the following geometrical interpretation. There exists a circle in the osculating plane tangent to {{math|'''γ'''(''s'')}} whose [[Taylor series]] to second order at the point of contact agrees with that of {{math|'''γ'''(''s'')}}. This is the [[osculating circle]] to the curve. The radius of the circle {{math|''R''(''s'')}} is called the [[radius of curvature]], and the curvature is the reciprocal of the radius of curvature:

:<math>\kappa(s) = \frac{1}{R(s)}.</math>

The tangent, curvature, and normal vector together describe the second-order behavior of a curve near a point. In three dimensions, the third-order behavior of a curve is described by a related notion of [[torsion of a curve|torsion]], which measures the extent to which a curve tends to move as a helical path in space. The torsion and curvature are related by the [[Frenet–Serret formulas]] (in three dimensions) and [[differential geometry of curves|their generalization]] (in higher dimensions).

===General expressions===
For a parametrically-defined space curve in three dimensions given in Cartesian coordinates by {{math|'''γ'''(''t'') {{=}} (''x''(''t''), ''y''(''t''), ''z''(''t''))}}, the curvature is

:<math>
\kappa=\frac{\sqrt{\bigl(z''y'-y''z'\bigr)\vphantom{'}^2+\bigl(x''z'-z''x'\bigr)\vphantom{'}^2+\bigl(y''x'-x''y'\bigr)\vphantom{'}^2}}
  {\bigl({x'}^2+{y'}^2+{z'}^2\bigr)\vphantom{'}^{3/2}},
</math>

where the prime denotes differentiation with respect to the parameter {{mvar|t}}. This can be expressed independently of the coordinate system by means of the formula<ref>A proof of this can be found at [https://mathworld.wolfram.com/Curvature.html the article on curvature] at [[Wolfram MathWorld]].</ref>

:<math>\kappa = \frac{\bigl\|\boldsymbol{\gamma}' \times \boldsymbol{\gamma}''\bigr\|}{\bigl\|\boldsymbol{\gamma}'\bigr\|\vphantom{'}^3}</math>

where × denotes the [[cross product|vector cross product]]. The following formula is valid for the curvature of curves in a Euclidean space of any dimension:

:<math>
\kappa = \frac{\sqrt{ \bigl\|\boldsymbol{\gamma}'\bigr\|\vphantom{'}^2 \bigl\|\boldsymbol{\gamma}''\bigr\|\vphantom{'}^2- \bigl(\boldsymbol{\gamma}'\cdot \boldsymbol{\gamma}''\bigr)\vphantom{'}^2 } }
{\bigl\|\boldsymbol{\gamma}'\bigr\|\vphantom{'}^3}.
</math>

===Curvature from arc and chord length===
Given two points {{mvar|P}} and {{mvar|Q}} on {{mvar|C}}, let {{math|''s''(''P'',''Q'')}} be the arc length of the portion of the curve between {{mvar|P}} and {{mvar|Q}} and let {{math|''d''(''P'',''Q'')}} denote the length of the line segment from {{mvar|P}} to {{mvar|Q}}. The curvature of {{mvar|C}} at {{mvar|P}} is given by the limit{{citation needed|date=December 2010}}

:<math>\kappa(P) = \lim_{Q\to P}\sqrt\frac{24\bigl(s(P,Q)-d(P,Q)\bigr)}{s(P,Q)\vphantom{Q}^3}</math>

where the limit is taken as the point {{mvar|Q}} approaches {{mvar|P}} on {{mvar|C}}. The denominator can equally well be taken to be {{math|''d''(''P'',''Q'')<sup>3</sup>}}. The formula is valid in any dimension. Furthermore, by considering the limit independently on either side of {{mvar|P}}, this definition of the curvature can sometimes accommodate a singularity at {{mvar|P}}. The formula follows by verifying it for the osculating circle.