Editing Differentiable curve (section)

== Frenet frame ==
{{main|Frenet–Serret formulas}}
[[File:Frenet frame.png|thumb|right|An illustration of the Frenet frame for a point on a space curve. {{math|''T''}} is the unit tangent, {{math|''P''}} the unit normal, and {{math|''B''}} the unit binormal.]]

A Frenet frame is a [[Moving frame|moving reference frame]] of {{math|''n''}} [[orthonormal]] vectors {{math|''e''<sub>''i''</sub>(''t'')}} which are used to describe a curve locally at each point {{math|'''γ'''(''t'')}}. It is the main tool in the differential geometric treatment of curves because it is far easier and more natural to describe local properties (e.g. curvature, torsion) in terms of a local reference system than using a global one such as Euclidean coordinates.

Given a {{math|''C''<sup>''n'' + 1</sup>}}-curve {{math|'''''γ'''''}} in <math>\mathbb{R}^n</math> which is regular of order {{math|''n''}} the Frenet frame for the curve is the set of orthonormal vectors
<math display="block">\mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)</math>
called [[Frenet–Serret formulas|Frenet vectors]]. They are constructed from the derivatives of {{math|'''''γ'''''(''t'')}} using the [[Gram–Schmidt process|Gram–Schmidt orthogonalization algorithm]] with
<math display="block">\begin{align}
\mathbf{e}_1(t) &= \frac{\boldsymbol{\gamma}'(t)}{\left\| \boldsymbol{\gamma}'(t) \right\|} \\[1ex]
\mathbf{e}_{j}(t) &= \frac{\overline{\mathbf{e}_{j}}(t)}{\left\|\overline{\mathbf{e}_{j}}(t) \right\|}, &
\overline{\mathbf{e}_{j}}(t) &= \boldsymbol{\gamma}^{(j)}(t) - \sum _{i=1}^{j-1} \left\langle \boldsymbol{\gamma}^{(j)}(t), \, \mathbf{e}_i(t) \right\rangle \, \mathbf{e}_i(t)
\vphantom{\Bigg\langle}
\end{align}</math>

The real-valued functions {{math|''χ''<sub>''i''</sub>(''t'')}} are called generalized curvatures and are defined as
<math display="block">\chi_i(t) = \frac{\bigl\langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \bigr\rangle}{\left\| \boldsymbol{\gamma}^'(t) \right\|} </math>

The Frenet frame and the generalized curvatures are invariant under reparametrization and are therefore differential geometric properties of the curve. For curves in <math>\mathbb R^3</math> <math>\chi_1(t)</math> is the curvature and <math>\chi_2(t)</math> is the torsion.

===Bertrand curve===
A '''Bertrand curve''' is a regular curve in <math>\mathbb R^3</math> with the additional property that there is a second curve in <math>\mathbb R^3</math> such that the [[#Normal vector or curvature vector|principal normal vectors]] to these two curves are identical at each corresponding point. In other words, if {{math|'''γ'''<sub>1</sub>(''t'')}} and {{math|'''γ'''<sub>2</sub>(''t'')}} are two curves in <math>\mathbb R^3</math> such that for any {{mvar|t}}, the two principal normals {{math|'''N'''<sub>1</sub>(''t''), '''N'''<sub>2</sub>(t)}} are equal, then {{math|'''γ'''<sub>1</sub>}} and {{math|'''γ'''<sub>2</sub>}} are Bertrand curves, and {{math|'''γ'''<sub>2</sub>}} is called the Bertrand mate of {{math|'''γ'''<sub>1</sub>}}. We can write {{math|'''γ'''<sub>2</sub>(''t'') {{=}}  '''γ'''<sub>1</sub>(''t'') + ''r'' '''N'''<sub>1</sub>(''t'')}} for some constant {{math|''r''}}.<ref name="do Carmo">{{cite book | last = do Carmo|first =Manfredo P. |author-link=Manfredo do Carmo | title=Differential Geometry of Curves and Surfaces | edition=revised & updated 2nd|publisher=Dover Publications, Inc. | year=2016|location=Mineola, NY | isbn=978-0-486-80699-0| pages=27–28}}</ref>

According to problem 25 in Kühnel's "Differential Geometry Curves – Surfaces – Manifolds", it is also true that two Bertrand curves that do not lie in the same two-dimensional plane are characterized by the existence of a linear relation {{math|''a'' ''κ''(''t'') + ''b'' ''τ''(''t'') {{=}} 1}} where {{math|''κ''(''t'')}} and {{math|''τ''(''t'')}}  are the curvature and torsion of {{math|'''γ'''<sub>1</sub>(''t'')}} and {{mvar|a}} and {{mvar|b}} are real constants with {{math|''a'' ≠ 0}}.<ref>{{cite book |page=53 |title=Differential Geometry: Curves, Surfaces, Manifolds |first=Wolfgang |last=Kühnel |location=Providence |publisher=AMS |year=2005 |isbn=0-8218-3988-8 }}</ref> Furthermore, the product of [[#Torsion|torsion]]s of a Bertrand pair of curves is constant.<ref>{{Cite web|url=https://mathworld.wolfram.com/BertrandCurves.html|title=Bertrand Curves|first=Eric W.|last=Weisstein|website=mathworld.wolfram.com}}</ref>
If {{math|'''γ'''<sub>1</sub>}} has more than one Bertrand mate then it has infinitely many. This only occurs when {{math|'''γ'''<sub>1</sub>}} is a circular helix.<ref name="do Carmo"/>