Editing Differentiable curve

{{short description|Study of curves from a differential point of view}}
{{About|curves in Euclidean space|curves in an arbitrary topological space|Curve}}

'''Differential geometry of curves''' is the branch of [[geometry]] that deals with [[smoothness (mathematics)|smooth]] [[curve]]s in the [[Euclidean plane|plane]] and the [[Euclidean space]] by methods of [[differential calculus|differential]] and [[integral calculus]].

Many [[list of curves|specific curves]] have been thoroughly investigated using the [[Synthetic geometry|synthetic approach]]. [[Differential geometry]] takes another path: curves are represented in a [[parametric equation|parametrized form]], and their geometric properties and various quantities associated with them, such as the [[curvature]] and the [[arc length]], are expressed via [[derivative]]s and [[integral]]s using [[vector calculus]]. One of the most important tools used to analyze a curve is the [[Frenet frame]], a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point.

The theory of curves is much simpler and narrower in scope than the [[differential geometry of surfaces|theory of surfaces]] and its higher-dimensional generalizations because a regular curve in a Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by the arc length (the ''natural parametrization''). From the point of view of a [[test particle|theoretical point particle]] on the curve that does not know anything about the ambient space, all curves would appear the same. Different space curves are only distinguished by how they bend and twist. Quantitatively, this is measured by the differential-geometric invariants called the ''[[curvature]]'' and the ''[[torsion of curves|torsion]]'' of a curve. The [[fundamental theorem of curves]] asserts that the knowledge of these invariants completely determines the curve.

== Definitions ==
{{main|Curve}}
A ''parametric'' {{math|''C''<sup>''r''</sup>}}-''curve'' or a {{math|''C''<sup>''r''</sup>}}-''parametrization'' is a [[vector-valued function]] 
<math display="block">\gamma: I \to \mathbb{R}^{n}</math>
that is {{mvar|r}}-times [[continuously differentiable]] (that is, the component functions of {{mvar|γ}} are continuously differentiable), where <math>n \isin \mathbb{N}</math>, <math>r \isin \mathbb{N} \cup \{\infty\}</math>, and {{mvar|I}} is a non-empty [[Interval (mathematics)|interval]] of real numbers. The {{em|image}} of the parametric curve is <math>\gamma[I] \subseteq \mathbb{R}^n</math>. The parametric curve {{mvar|γ}} and its image {{math|''γ''[''I'']}} must be distinguished because a given subset of <math>\mathbb{R}^n</math> can be the image of many distinct parametric curves. The parameter {{mvar|t}} in {{math|''γ''(''t'')}} can be thought of as representing time, and {{mvar|γ}} the [[trajectory]] of a moving point in space. When {{mvar|I}} is a closed interval {{math|[''a'',''b'']}}, {{math|''γ''(''a'')}} is called the starting point and {{math|''γ''(''b'')}} is the endpoint of {{mvar|γ}}. If the starting and the end points coincide (that is, {{math|''γ''(''a'') {{=}} ''γ''(''b'')}}), then {{mvar|γ}} is a ''closed curve'' or a ''loop''. To be a {{math|''C''<sup>''r''</sup>}}-loop, the function {{mvar|γ}} must be {{mvar|r}}-times continuously differentiable and satisfy {{math|''γ''<sup>(''k'')</sup>(''a'') {{=}} ''γ''<sup>(''k'')</sup>(''b'')}} for {{math|0 ≤ ''k'' ≤ ''r''}}.

The parametric curve is {{em|simple}} if
<math display="block"> \gamma|_{(a,b)}: (a,b) \to \mathbb{R}^{n} </math>
is [[injective]]. It is {{em|analytic}} if each component function of {{mvar|γ}} is an [[analytic function]], that is, it is of class {{math|''C''<sup>''ω''</sup>}}.

The curve {{mvar|γ}} is ''regular of order'' {{mvar|m}} (where {{math|''m'' ≤ ''r''}}) if, for every {{math|''t'' ∈ ''I''}},
<math display="block">\left\{ \gamma'(t),\gamma''(t),\ldots,{\gamma^{(m)}}(t) \right\}</math>
is a [[linearly independent]] subset of <math>\mathbb{R}^n</math>. In particular, a parametric {{math|''C''<sup>1</sup>}}-curve {{mvar|γ}} is {{em|regular}} if and only if {{math|''γ''{{prime}}(''t'') ≠ '''0'''}} for any {{math|''t'' ∈ ''I''}}.

== Re-parametrization and equivalence relation ==
{{See also|Position vector|Vector-valued function}}

Given the image of a parametric curve, there are several different parametrizations of the parametric curve. Differential geometry aims to describe the properties of parametric curves that are invariant under certain reparametrizations. A suitable [[equivalence relation]] on the set of all parametric curves must be defined. The differential-geometric properties of a parametric curve (such as its length, its [[#Frenet frame|Frenet frame]], and its generalized curvature) are invariant under reparametrization and therefore properties of the [[equivalence class]] itself. The equivalence classes are called {{math|''C''<sup>''r''</sup>}}-curves and are central objects studied in the differential geometry of curves.

Two parametric {{math|''C''<sup>''r''</sup>}}-curves, <math>\gamma_1 : I_1 \to \mathbb{R}^n</math> and <math>\gamma_2 : I_2 \to \mathbb{R}^n</math>, are said to be {{em|equivalent}} if and only if there exists a [[bijective]] {{math|''C''<sup>''r''</sup>}}-map {{math|''φ'' : ''I''<sub>1</sub> → ''I''<sub>2</sub>}} such that
<math display="block">\forall t \in I_1: \quad \varphi'(t) \neq 0</math>
and
<math display="block">\forall t \in I_1: \quad \gamma_2\bigl(\varphi(t)\bigr) = \gamma_1(t).</math>
{{math|''γ''<sub>2</sub>}} is then said to be a {{em|re-parametrization}} of {{math|''γ''<sub>1</sub>}}.

Re-parametrization defines an equivalence relation on the set of all parametric {{math|''C''<sup>''r''</sup>}}-curves of class {{math|''C''<sup>''r''</sup>}}. The equivalence class of this relation simply a {{math|''C''<sup>''r''</sup>}}-curve.

An even ''finer'' equivalence relation of oriented parametric {{math|''C''<sup>''r''</sup>}}-curves can be defined by requiring {{mvar|φ}} to satisfy {{math|''φ''{{prime}}(''t'') > 0}}.

Equivalent parametric {{math|''C''<sup>''r''</sup>}}-curves have the same image, and equivalent oriented parametric {{math|''C''<sup>''r''</sup>}}-curves even traverse the image in the same direction.

==Length and natural parametrization{{anchor|Length|Natural parametrization}}==
{{main|Arc length}}
{{see also|Curve#Length of a curve}}

The length {{mvar|l}} of a parametric {{math|''C''<sup>1</sup>}}-curve <math>\gamma : [a, b] \to \mathbb{R}^n</math> is defined as
<math display="block">l ~ \stackrel{\text{def}}{=} ~ \int_a^b \left\| \gamma'(t) \right\| \, \mathrm{d}{t}.</math>
The length of a parametric curve is invariant under reparametrization and is therefore a differential-geometric property of the parametric curve.

For each regular parametric {{math|''C''<sup>''r''</sup>}}-curve <math>\gamma : [a, b] \to \mathbb{R}^n</math>, where {{math|''r'' ≥ 1}}, the function is defined
<math display="block">\forall t \in [a,b]: \quad s(t) ~ \stackrel{\text{def}}{=} ~ \int_a^t \left\| \gamma'(x) \right\| \, \mathrm{d}{x}.</math>
Writing {{math|''{{overline|γ}}''(s) {{=}} ''γ''(''t''(''s''))}}, where {{math|''t''(''s'')}} is the inverse function of {{math|''s''(''t'')}}. This is a re-parametrization {{math|''{{overline|γ}}''}} of {{mvar|γ}} that is called an ''{{vanchor|arc-length parametrization}}'',  ''natural parametrization'', ''unit-speed parametrization''. The parameter {{math|''s''(''t'')}} is called the {{em|natural parameter}} of {{mvar|γ}}.

This parametrization is preferred because the natural parameter {{math|''s''(''t'')}} traverses the image of {{mvar|γ}} at unit speed, so that
<math display="block">\forall t \in I: \quad \left\| \overline{\gamma}'\bigl(s(t)\bigr) \right\| = 1.</math>
In practice, it is often very difficult to calculate the natural parametrization of a parametric curve, but it is useful for theoretical arguments.

For a given parametric curve {{mvar|γ}}, the natural parametrization is unique up to a shift of parameter.

The quantity
<math display="block">E(\gamma) ~ \stackrel{\text{def}}{=} ~ \frac{1}{2} \int_a^b \left\| \gamma'(t) \right\|^2 ~ \mathrm{d}{t}</math>
is sometimes called the {{em|energy}} or [[action (physics)|action]] of the curve; this name is justified because the [[geodesic]] equations are the [[Euler–Lagrange equation]]s of motion for this action.

== 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"/>

== 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>

== Main theorem of curve theory ==
{{main|Fundamental theorem of curves}}
Given {{math|''n'' − 1}} functions:
<math display="block">\chi_i \in C^{n-i}([a,b],\mathbb{R}^n) , \quad \chi_i(t) > 0 ,\quad  1 \leq i \leq n-1</math>
then there exists a unique (up to transformations using the [[Euclidean group]]) {{math|''C''<sup>''n'' + 1</sup>}}-curve {{math|''γ''}} which is regular of order {{mvar|n}} and has the following properties:
<math display="block">\begin{align}
\|\gamma'(t)\| &= 1  & t \in [a,b] \\
\chi_i(t) &= \frac{ \langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \boldsymbol{\gamma}'(t) \|}
\end{align}</math>
where the set
<math display="block">\mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)</math>
is the Frenet frame for the curve.

By additionally providing a start {{math|''t''<sub>0</sub>}} in {{math|''I''}}, a starting point {{math|''p''<sub>0</sub>}} in <math>\mathbb{R}^n</math> and an initial positive orthonormal Frenet frame {{math|{{mset|''e''<sub>1</sub>, ..., ''e''<sub>''n'' − 1</sub>}}}} with
<math display="block">\begin{align}
\boldsymbol{\gamma}(t_0) &= \mathbf{p}_0 \\
\mathbf{e}_i(t_0) &= \mathbf{e}_i ,\quad  1 \leq i \leq n-1
\end{align}</math>
the Euclidean transformations are eliminated to obtain a unique curve {{math|''γ''}}.

== Frenet–Serret formulas ==

{{main|Frenet–Serret formulas}}

The Frenet–Serret formulas are a set of [[ordinary differential equations]] of first order. The solution is the set of Frenet vectors describing the curve specified by the generalized curvature functions {{math|''χ''<sub>''i''</sub>}}.

=== 2 dimensions ===

<math display="block"> 
\begin{bmatrix}
 \mathbf{e}_1'(t) \\ \mathbf{e}_2'(t)
\end{bmatrix}

=

\left\Vert \gamma'(t) \right\Vert

\begin{bmatrix}
         0  & \kappa(t) \\
 -\kappa(t) &         0 \\
\end{bmatrix}

\begin{bmatrix}
\mathbf{e}_1(t) \\ \mathbf{e}_2(t)
\end{bmatrix} 
</math>

=== 3 dimensions ===

<math display="block"> 
\begin{bmatrix}
 \mathbf{e}_1'(t) \\[0.75ex]
 \mathbf{e}_2'(t) \\[0.75ex]
 \mathbf{e}_3'(t)
\end{bmatrix}

=

\left\Vert \gamma'(t) \right\Vert

\begin{bmatrix}
          0 &  \kappa(t) &       0 \\[1ex]
 -\kappa(t) &          0 & \tau(t) \\[1ex]
          0 &   -\tau(t) &       0
\end{bmatrix}

\begin{bmatrix}
 \mathbf{e}_1(t) \\[1ex]
 \mathbf{e}_2(t) \\[1ex]
 \mathbf{e}_3(t)
\end{bmatrix} 
</math>

=== {{math|''n''}} dimensions (general formula) ===

<math display="block"> 
\begin{bmatrix}
 \mathbf{e}_1'(t) \\[1ex]
 \mathbf{e}_2'(t) \\[1ex]
           \vdots \\[1ex]
 \mathbf{e}_{n-1}'(t) \\[1ex]
 \mathbf{e}_n'(t) \\[1ex]
\end{bmatrix}

=

\left\Vert \gamma'(t) \right\Vert

\begin{bmatrix}
          0 &  \chi_1(t) & \cdots &              0 &             0 \\[1ex]
 -\chi_1(t) &          0 & \cdots &              0 &             0 \\[1ex]
     \vdots &     \vdots & \ddots &         \vdots &        \vdots \\[1ex]
          0 &          0 & \cdots &              0 & \chi_{n-1}(t) \\[1ex]
          0 &          0 & \cdots & -\chi_{n-1}(t) &             0 \\[1ex]
\end{bmatrix}

\begin{bmatrix}
 \mathbf{e}_1(t) \\[1ex]
 \mathbf{e}_2(t) \\[1ex]
          \vdots \\[1ex]
 \mathbf{e}_{n-1}(t) \\[1ex]
 \mathbf{e}_n(t) \\[1ex]
\end{bmatrix} 
</math>

==See also==
*[[List of curves topics]]

==References==
{{Reflist}}

==Further reading==
*{{cite book |first=Erwin |last=Kreyszig |title=Differential Geometry |publisher=Dover Publications |location=New York |year=1991 |isbn=0-486-66721-9 }} Chapter II is a classical treatment of ''Theory of Curves'' in 3-dimensions.

{{Differential transforms of plane curves}}
{{Curvature}}
{{tensors}}

[[Category:Differential geometry]]
[[Category:Curves]]