Editing Frenet–Serret formulas (section)

==Definitions==
[[Image:FrenetTN.svg|thumb|right|350px|
{{legend-line|solid black 2px|The {{math|'''T'''}} and {{math|'''N'''}} vectors at two points on a plane curve}}
{{legend-line|dotted black 2px|A translated version of the second frame.}}
{{legend-line|dashed black 2px|The change in {{math|'''T''': δ'''T''''}}.}} 
{{mvar|δs}} is the distance between the points. In the limit <math>\tfrac{d\mathbf{T}}{ds}</math> will be in the direction {{math|'''N'''}} and the curvature describes the speed of rotation of the frame.]]

Let {{math|'''r'''(''t'')}} be a [[curve]] in [[Euclidean space]], representing the [[position vector]] of the particle as a function of time. The Frenet–Serret formulas apply to curves which are ''non-degenerate'', which roughly means that they have nonzero [[curvature]]. More formally, in this situation the [[velocity]] vector {{math|'''r'''&prime;(''t'')}} and the [[acceleration]] vector {{math|'''r'''&prime;&prime;(''t'')}} are required not to be proportional.

Let {{math|''s''(''t'')}} represent the [[arc length]] which the particle has moved along the [[curve]] in time {{mvar|t}}. The quantity {{mvar|s}} is used to give the curve traced out by the trajectory of the particle a [[Rectifiable path|natural parametrization]] by arc length (i.e. [[Differentiable curve#Length and natural parametrization|arc-length parametrization]]), since many different particle paths may trace out the same geometrical curve by traversing it at different rates. In detail, {{mvar|s}} is given by
<math display=block>s(t) = \int_0^t \left\|\mathbf{r}'(\sigma)\right\|d\sigma.</math>
Moreover, since we have assumed that {{math|'''r'''&prime; ≠ 0}}, it follows that {{math|''s''(''t'')}} is a strictly monotonically increasing function. Therefore, it is possible to solve for {{mvar|t}} as a function of {{mvar|s}}, and thus to write {{math|1='''r'''(''s'') = '''r'''(''t''(''s''))}}. The curve is thus parametrized in a preferred manner by its arc length.

With a non-degenerate curve {{math|'''r'''(''s'')}}, parameterized by its arc length, it is now possible to define the '''Frenet–Serret frame''' (or '''{{math|TNB}} frame'''):
<ul>
<li>
The tangent unit vector {{math|'''T'''}} is defined as

<math display="block"> \mathbf{T} := \frac{ \mathrm{d} \mathbf{r} }{ \mathrm{d} s } .</math>
</li>
<li>
The normal unit vector {{math|'''N'''}} is defined as

<math display="block"> \mathbf{N} :=
{
  \frac{ \mathrm{d} \mathbf{T} }{ \mathrm{d} s } \over
  \left\|
	\frac{ \mathrm{d} \mathbf{T}}{ \mathrm{d} s}
  \right\|
},</math>
from which it follows, since {{math|'''T'''}} always has unit [[magnitude (mathematics)|magnitude]], that {{math|'''N'''}} (the change of {{math|'''T'''}}) is always perpendicular to {{math|'''T'''}}, since there is no change in length of {{math|'''T'''}}. Note that by calling curvature <math> \kappa = \left\| \frac{ \mathrm{d} \mathbf{T}}{ \mathrm{d} s}\right\|</math> we automatically obtain the first relation.
</li>
</ul><ul>
<li>
The binormal unit vector {{math|'''B'''}} is defined as the [[cross product]] of {{math|'''T'''}} and {{math|'''N'''}}:

<math display="block"> \mathbf{B} := \mathbf{T} \times \mathbf{N},</math>
</li>
</ul>

[[Image:frenetframehelix.gif|thumb|right|400px|The Frenet–Serret frame moving along a [[helix]]. The {{math|'''T'''}} is represented by the blue arrow, {{math|'''N'''}} is represented by the red arrow while {{math|'''B'''}} is represented by the black arrow.]]

from which it follows that {{math|'''B'''}} is always perpendicular to both {{math|'''T'''}} and {{math|'''N'''}}. Thus, the three unit vectors {{math|'''T''', '''N''', '''B'''}} are all perpendicular to each other.

The '''Frenet–Serret formulas''' are:

<math display=block>
\begin{align}
\frac{\mathrm{d} \mathbf{T} }{ \mathrm{d} s } &= \kappa\mathbf{N}, \\[4pt]
\frac{\mathrm{d} \mathbf{N} }{ \mathrm{d} s } &= -\kappa\mathbf{T} + \tau\mathbf{B},\\[4pt]
\frac{\mathrm{d} \mathbf{B} }{ \mathrm{d} s } &= -\tau\mathbf{N},
\end{align}
</math>

where {{mvar|&kappa;}} is the [[curvature]] and {{mvar|&tau;}} is the [[Torsion of curves|torsion]].

The Frenet–Serret formulas are also known as ''Frenet–Serret theorem'', and can be stated more concisely using matrix notation:<ref>{{harvnb|Kühnel|2002|loc=§1.9}}</ref>
<math display=block> \begin{bmatrix} \mathbf{T'} \\ \mathbf{N'} \\ \mathbf{B'} \end{bmatrix} = \begin{bmatrix}
 0 & \kappa & 0 \\
 -\kappa & 0 & \tau \\
 0 & -\tau & 0
\end{bmatrix}
\begin{bmatrix} \mathbf{T} \\ \mathbf{N} \\ \mathbf{B} \end{bmatrix}.</math>

This matrix is [[Skew-symmetric matrix|skew-symmetric]].