Editing Differentiable curve (section)

== Special Frenet vectors and generalized curvatures ==
{{main|Frenet–Serret formulas}}
The first three Frenet vectors and generalized curvatures can be visualized in three-dimensional space. They have additional names and more semantic information attached to them.

=== Tangent vector ===

If a curve {{math|'''γ'''}} represents the path of a particle over time, then the instantaneous [[velocity]] of the particle at a given position {{math|''P''}} is expressed by a [[Vector (geometric)|vector]], called the ''[[tangent vector]]'' to the curve at {{math|''P''}}. Mathematically, given a parametrized {{math|''C''<sup>1</sup>}} curve {{math|1='''''γ''''' = '''''γ'''''(''t'')}}, for every value {{math|''t'' {{=}} ''t''<sub>0</sub>}} of the time parameter, the vector
<math display="block"> \boldsymbol{\gamma}'(t_0) = \left.\frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{\gamma}(t)\right|_{t=t_0} </math>
is the tangent vector at the point {{math|''P'' {{=}} '''γ'''(''t''<sub>0</sub>)}}. Generally speaking, the tangent vector may be [[zero vector|zero]]. The tangent vector's magnitude
<math display="block">\left\|\boldsymbol{\gamma}'(t_0)\right\|</math>
is the speed at the time {{math|''t''<sub>0</sub>}}.

The first Frenet vector {{math|'''e'''<sub>1</sub>(''t'')}} is the unit tangent vector in the same direction, called simply the tangent direction, defined at each regular point of {{math|'''γ'''}}:
<math display="block">\mathbf{e}_{1}(t) = \frac{ \boldsymbol{\gamma}'(t) }{ \left\| \boldsymbol{\gamma}'(t) \right\|}.</math>
If the time parameter is replaced by the arc length, {{math|''t'' {{=}} ''s''}}, then the tangent vector has unit length and the formula simplifies:
<math display="block">\mathbf{e}_{1}(s) = \boldsymbol{\gamma}'(s).</math>
However, then it is no longer applicable the interpretation in terms of the particle's velocity (with [[dimension (physics)|dimension]] of length per time).
The tangent direction determines the orientation of the curve, or the forward direction, corresponding to the increasing values of the parameter. The tangent direction taken as a curve traces the [[spherical image]] of the original curve.

=== Normal vector or curvature vector ===

A curve ''[[normal vector]]'', sometimes called the '''curvature vector''', indicates the deviance of the curve from being a straight line.
It is defined as the [[vector rejection]] of the particle's [[acceleration]] from the tangent direction:
<math display="block">\overline{\mathbf{e}_2}(t) = \boldsymbol{\gamma}''(t) - \bigl\langle \boldsymbol{\gamma}''(t), \mathbf{e}_1(t) \bigr\rangle \, \mathbf{e}_1(t),</math>
where the acceleration is defined as the second derivative of position with respect to time:
<math display="block"> \boldsymbol{\gamma}''(t_0) = \left.\frac{\mathrm{d}^2}{\mathrm{d}t^2}\boldsymbol{\gamma}(t)\right|_{t=t_0} </math>

Its normalized form, the unit normal vector, is the second Frenet vector {{math|'''e'''<sub>2</sub>(''t'')}} and is defined as
<math display="block">\mathbf{e}_2(t) = \frac{\overline{\mathbf{e}_2}(t)} {\left\| \overline{\mathbf{e}_2}(t) \right\|}.</math>

The tangent and the normal vector at point {{math|''t''}} define the [[osculating plane]] at point {{math|''t''}}.

It can be shown that {{math|'''ē'''<sub>2</sub>(''t'') ∝ '''e'''{{prime}}<sub>1</sub>(''t'')}}. Therefore,
<math display="block">\mathbf{e}_2(t) = \frac{\mathbf{e}_1'(t)}{\left\| \mathbf{e}_1'(t) \right\|}.</math>

===Curvature===
{{main|Curvature of space curves}}

The first generalized curvature {{math|''χ''<sub>1</sub>(''t'')}} is called curvature and measures the deviance of {{math|''γ''}} from being a straight line relative to the osculating plane. It is defined as
<math display="block">\kappa(t) = \chi_1(t) = \frac{\bigl\langle \mathbf{e}_1'(t), \mathbf{e}_2(t) \bigr\rangle}{\left\| \boldsymbol{\gamma}'(t) \right\|}</math>
and is called the [[curvature]] of {{math|''γ''}} at point {{math|''t''}}. It can be shown that
<math display="block">\kappa(t) = \frac{\left\| \mathbf{e}_1'(t) \right\|}{\left\| \boldsymbol{\gamma}'(t) \right\|}.</math>

The [[Multiplicative inverse|reciprocal]] of the curvature
<math display="block">\frac{1}{\kappa(t)}</math>
is called the [[radius of curvature (mathematics)|radius of curvature]].

A circle with radius {{math|''r''}} has a constant curvature of 
<math display="block">\kappa(t) = \frac{1}{r}</math>
whereas a line has a curvature of 0.

=== Binormal vector ===
The unit binormal vector is the third Frenet vector {{math|'''e'''<sub>3</sub>(''t'')}}. It is always orthogonal to the unit tangent and normal vectors at {{math|''t''}}. It is defined as

<math display="block">\mathbf{e}_3(t) = \frac{\overline{\mathbf{e}_3}(t)} {\left\| \overline{\mathbf{e}_3}(t) \right\|}
,  \quad
\overline{\mathbf{e}_3}(t) = \boldsymbol{\gamma}'''(t) - \bigr\langle \boldsymbol{\gamma}'''(t), \mathbf{e}_1(t) \bigr\rangle \, \mathbf{e}_1(t)
- \bigl\langle \boldsymbol{\gamma}'''(t), \mathbf{e}_2(t) \bigr\rangle \,\mathbf{e}_2(t)
</math>

In 3-dimensional space, the equation simplifies to
<math display="block">\mathbf{e}_3(t) = \mathbf{e}_1(t) \times \mathbf{e}_2(t)</math>
or to 
<math display="block">\mathbf{e}_3(t) = -\mathbf{e}_1(t) \times \mathbf{e}_2(t),</math>
That either sign may occur is illustrated by the examples of a right-handed helix and a left-handed helix.

=== Torsion ===
{{main|Torsion of a curve}}

The second generalized curvature {{math|''χ''<sub>2</sub>(''t'')}} is called {{em|torsion}} and measures the deviance of {{math|''γ''}} from being a [[plane curve]]. In other words, if the torsion is zero, the curve lies completely in the same osculating plane (there is only one osculating plane for every point {{math|''t''}}). It is defined as
<math display="block">\tau(t) = \chi_2(t) = \frac{\bigl\langle \mathbf{e}_2'(t), \mathbf{e}_3(t) \bigr\rangle}{\left\| \boldsymbol{\gamma}'(t) \right\|}</math>
and is called the [[torsion (differential geometry)|torsion]] of {{math|''γ''}} at point {{math|''t''}}.

=== Aberrancy ===
The [[third derivative]] may be used to define '''aberrancy''', a metric of [[Circle|non-circularity]] of a curve.<ref>{{cite journal|last=Schot|first=Stephen|title=Aberrancy: Geometry of the Third Derivative|journal=Mathematics Magazine|date=November 1978|volume=51|series=5|issue=5|pages=259–275|jstor=2690245|doi=10.2307/2690245}}</ref><ref>{{cite journal | title=Measures of Aberrancy | journal=Real Analysis Exchange | publisher=Michigan State University Press | volume=32 | issue=1 | year=2007 | issn=0147-1937 | doi=10.14321/realanalexch.32.1.0233 | page=233| last1=Cameron Byerley | last2=Russell a. Gordon | doi-access=free }}</ref><ref>{{cite journal | last=Gordon | first=Russell A. | title=The aberrancy of plane curves | journal=The Mathematical Gazette | publisher=Cambridge University Press (CUP) | volume=89 | issue=516 | year=2004 | issn=0025-5572 | doi=10.1017/s0025557200178271 | pages=424–436| s2cid=118533002 }}</ref>