Editing Frenet–Serret formulas (section)

==Applications and interpretation==

=== Kinematics of the frame ===
[[Image:Frenet-Serret moving frame1.png|right|thumb|The Frenet–Serret frame moving along a [[helix]] in space]]

The Frenet–Serret frame consisting of the tangent {{math|'''T'''}}, normal {{math|'''N'''}}, and binormal {{math|'''B'''}} collectively forms an [[orthonormal basis]] of 3-space. At each point of the curve, this ''attaches'' a [[frame of reference]] or [[rectilinear grid|rectilinear]] [[coordinate system]] (see image).

The Frenet–Serret formulas admit a [[kinematics|kinematic]] interpretation. Imagine that an observer moves along the curve in time, using the attached frame at each point as their coordinate system. The Frenet–Serret formulas mean that this coordinate system is constantly rotating as an observer moves along the curve. Hence, this coordinate system is always [[Non-inertial reference frame|non-inertial]]. The [[angular momentum]] of the observer's coordinate system is proportional to the [[Darboux vector]] of the frame.

[[Image:TNB frame momenta.svg|left|thumb|A top whose axis is situated along the binormal is observed to rotate with angular speed {{mvar|&kappa;}}. If the axis is along the tangent, it is observed to rotate with angular speed {{mvar|&tau;}}.]]
Concretely, suppose that the observer carries an (inertial) [[Spinning top|top]] (or [[gyroscope]]) with them along the curve. If the axis of the top points along the tangent to the curve, then it will be observed to rotate about its axis with [[angular velocity]] −τ relative to the observer's non-inertial coordinate system. If, on the other hand, the axis of the top points in the binormal direction, then it is observed to rotate with angular velocity −κ. This is easily visualized in the case when the curvature is a positive constant and the torsion vanishes. The observer is then in [[uniform circular motion]]. If the top points in the direction of the binormal, then by [[conservation of angular momentum]] it must rotate in the ''opposite'' direction of the circular motion. In the limiting case when the curvature vanishes, the observer's normal [[precess]]es about the tangent vector, and similarly the top will rotate in the opposite direction of this precession.

The general case is illustrated [[#Illustrations|below]]. There are further [[commons:Category:Illustrations for curvature and torsion of curves|illustrations]] on Wikimedia.

==== Applications ====
The kinematics of the frame have many applications in the sciences.
*  In the [[life sciences]], particularly in models of microbial motion, considerations of the Frenet–Serret frame have been used to explain the mechanism by which a moving organism in a viscous medium changes its direction.<ref>Crenshaw (1993).</ref>
* In physics, the Frenet–Serret frame is useful when it is impossible or inconvenient to assign a natural coordinate system for a trajectory. Such is often the case, for instance, in [[relativity theory]]. Within this setting, Frenet–Serret frames have been used to model the precession of a gyroscope in a gravitational well.<ref>Iyer and Vishveshwara (1993).</ref><!--More elementary applications?  Classic papers on coriolis effects maybe?-->
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====Graphical Illustrations====

# Example of a moving Frenet basis ({{math|'''T'''}} in blue, {{math|'''N'''}} in green, {{math|'''B'''}} in purple) along [[Viviani's curve]].

[[File:Frenet-Serret-frame along Vivani-curve.gif]]

#<li value=2> On the example of a [[torus knot]], the tangent vector {{math|'''T'''}}, the normal vector {{math|'''N'''}}, and the binormal vector {{math|'''B'''}}, along with the curvature {{math|''κ''(''s'')}}, and the torsion {{math|''τ''(''s'')}} are displayed. <br> At the peaks of the torsion function the rotation of the Frenet–Serret frame {{math|('''T''','''N''','''B''')}} around the tangent vector is clearly visible.</li>

[[File:Torus-Knot nebeneinander animated.gif]]

#<li value=3> The kinematic significance of the curvature is best illustrated with plane curves (having constant torsion equal to zero). See the page on [[Curvature#Curvature of plane curves|curvature of plane curves]].</li>

=== Frenet–Serret formulas in calculus ===
The Frenet–Serret formulas are frequently introduced in courses on [[multivariable calculus]] as a companion to the study of space curves such as the [[helix]]. A helix can be characterized by the height {{math|2π''h''}} and radius {{mvar|r}} of a single turn. The curvature and torsion of a helix (with constant radius) are given by the formulas
<math display=block>\begin{align}
  \kappa &= \frac{r}{r^2+h^2} \\[4pt]
  \tau &= \pm\frac{h}{r^2+h^2}.
\end{align}</math>
[[Image:Frenet-Serret helices.png|right|thumb|Two helices (slinkies) in space. (a) A more compact helix with higher curvature and lower torsion. (b) A stretched out helix with slightly higher torsion but lower curvature.]]
The sign of the torsion is determined by the right-handed or left-handed [[right-hand rule|sense]] in which the helix twists around its central axis. Explicitly, the parametrization of a single turn of a right-handed helix with height {{math|2π''h''}} and radius {{mvar|r}} is
<math display=block>\begin{align}
  x &= r \cos t \\
  y &= r \sin t \\
  z &= ht \\
  (0 &\leq t \leq 2 \pi)
\end{align}</math>
and, for a left-handed helix,
<math display=block>\begin{align}
  x &= r \cos t \\
  y &= -r \sin t \\
  z &= ht \\
  (0 &\leq t \leq 2 \pi).
\end{align}</math>
Note that these are not the arc length parametrizations (in which case, each of {{math|''x'', ''y'', ''z''}} would need to be divided by <math>\sqrt{h^2+r^2}</math>.)

In his expository writings on the geometry of curves, [[Rudy Rucker]]<ref name = rucker>{{cite web|last = Rucker |first = Rudy|date = 1999|title = Watching Flies Fly: Kappatau Space Curves |publisher = San Jose State University|url = http://www.mathcs.sjsu.edu/faculty/rucker/kaptaudoc/ktpaper.htm|url-status = dead |archive-url =https://web.archive.org/web/20041015020304/http://www.mathcs.sjsu.edu/faculty/rucker/kaptaudoc/ktpaper.htm|archive-date = 15 October 2004}}</ref> employs the model of a [[slinky]] to explain the meaning of the torsion and curvature. The slinky, he says, is characterized by the property that the quantity
<math display=block> A^2 = h^2+r^2</math>
remains constant if the slinky is vertically stretched out along its central axis. (Here {{math|2π''h''}} is the height of a single twist of the slinky, and {{mvar|r}} the radius.) In particular, curvature and torsion are complementary in the sense that the torsion can be increased at the expense of curvature by stretching out the slinky.

=== Taylor expansion ===
Repeatedly differentiating the curve and applying the Frenet–Serret formulas gives the following [[Taylor's theorem|Taylor approximation]] to the curve near {{math|1=''s'' = 0}} if the curve is parameterized by arclength:<ref>{{harvnb|Kühnel|2002|p=19}}</ref>
<math display=block>\mathbf r(s) = \mathbf r(0) + \left(s-\frac{s^3\kappa^2(0)}{6}\right)\mathbf T(0) + \left(\frac{s^2\kappa(0)}{2}+\frac{s^3\kappa'(0)}{6}\right)\mathbf N(0) + \left(\frac{s^3\kappa(0)\tau(0)}{6}\right)\mathbf B(0) + o(s^3).</math>

For a generic curve with nonvanishing torsion, the projection of the curve onto various coordinate planes in the {{math|'''T''', '''N''', '''B'''}} coordinate system at {{nowrap|1=''s'' = 0}} have the following interpretations:

*The ''[[osculating plane]]'' is the plane [[linear span|containing]] {{math|'''T'''}} and {{math|'''N'''}}. The projection of the curve onto this plane has the form:<math display="block">
  \mathbf r(0) + s\mathbf T(0) + \frac{s^2\kappa(0)}{2} \mathbf N(0) + o(s^2).
</math>This is a [[parabola]] up to terms of order {{math|''O''(''s''<sup>2</sup>)}}, whose curvature at 0 is equal to {{math|''κ''(0)}}. The osculating plane has the special property that the distance from the curve to the osculating plane is {{math|''O''(''s''<sup>3</sup>)}}, while the distance from the curve to any other plane is no better than {{math|''O''(''s''<sup>2</sup>)}}. This can be seen from the above Taylor expansion. Thus in a sense the osculating plane is the closest plane to the curve at a given point.
*The ''[[Normal plane (geometry)|normal plane]]'' is the plane containing {{math|'''N'''}} and {{math|'''B'''}}. The projection of the curve onto this plane has the form:<math display="block">
  \mathbf r(0) + \left(\frac{s^2\kappa(0)}{2}+\frac{s^3\kappa'(0)}{6}\right)\mathbf N(0) + \left(\frac{s^3\kappa(0)\tau(0)}{6}\right)\mathbf B(0)+ o(s^3)
</math>which is a [[cuspidal cubic]] to order {{math|''o''(''s''<sup>3</sup>)}}.
*The '''rectifying plane''' is the plane containing {{math|'''T'''}} and {{math|'''B'''}}. The projection of the curve onto this plane is:<math display="block">
  \mathbf r(0) + \left(s-\frac{s^3\kappa^2(0)}{6}\right)\mathbf T(0) + \left(\frac{s^3\kappa(0)\tau(0)}{6}\right)\mathbf B(0)+ o(s^3)
</math>which traces out the graph of a [[cubic polynomial]] to order {{math|''o''(''s''<sup>3</sup>)}}.

=== Ribbons and tubes ===
[[File:Ribbon-Frenet.png|thumb|350px|A ribbon defined by a curve of constant torsion and a highly oscillating curvature. The arc length parameterization of the curve was defined via integration of the Frenet–Serret equations.]]
The Frenet–Serret apparatus allows one to define certain optimal ''ribbons'' and ''tubes'' centered around a curve. These have diverse applications in [[materials science]] and [[elasticity theory]],<ref>Goriely ''et al.'' (2006).</ref> as well as to [[computer graphics]].<ref>Hanson.</ref>

The '''Frenet ribbon'''<ref>For terminology, see {{cite book |last=Sternberg |date=1964 |title=Lectures on Differential Geometry |url=https://archive.org/details/lecturesondiffer00ster_853|url-access=limited |publisher=Englewood Cliffs, N.J., Prentice-Hall |page=[https://archive.org/details/lecturesondiffer00ster_853/page/n263 252]-254|isbn=9780135271506 }}.</ref> along a curve {{mvar|C}} is the surface traced out by sweeping the line segment {{math|[&minus;'''N''','''N''']}} generated by the unit normal along the curve. This surface is sometimes confused with the [[tangent developable]], which is the [[envelope (mathematics)|envelope]] {{mvar|E}} of the osculating planes of {{mvar|C}}. This is perhaps because both the Frenet ribbon and {{mvar|E}} exhibit similar properties along {{mvar|C}}. Namely, the tangent planes of both sheets of {{mvar|E}}, near the singular locus {{mvar|C}} where these sheets intersect, approach the osculating planes of {{mvar|C}}; the tangent planes of the Frenet ribbon along {{mvar|C}} are equal to these osculating planes. The Frenet ribbon is in general not developable.

=== Congruence of curves ===
In classical [[Euclidean geometry]], one is interested in studying the properties of figures in the plane which are ''invariant'' under congruence, so that if two figures are congruent then they must have the same properties. The Frenet–Serret apparatus presents the curvature and torsion as numerical invariants of a space curve.

Roughly speaking, two curves {{mvar|C}} and {{mvar|C'}} in space are ''congruent'' if one can be rigidly moved to the other. A rigid motion consists of a combination of a translation and a rotation. A translation moves one point of {{mvar|C}} to a point of {{mvar|C'}}. The rotation then adjusts the orientation of the curve {{mvar|C}} to line up with that of {{mvar|C'}}. Such a combination of translation and rotation is called a [[Euclidean transformation|Euclidean motion]]. In terms of the parametrization {{math|'''r'''(''t'')}} defining the first curve {{mvar|C}}, a general Euclidean motion of {{mvar|C}} is a composite of the following operations:
* (''Translation'')  {{math|'''r'''(''t'') → '''r'''(''t'') + '''v'''}}, where {{math|'''v'''}} is a constant vector.
* (''Rotation'')  {{math|'''r'''(''t'') + '''v''' → ''M''('''r'''(''t'') + '''v''')}}, where {{mvar|M}} is the matrix of a rotation.

The Frenet–Serret frame is particularly well-behaved with regard to Euclidean motions. First, since {{math|'''T'''}}, {{math|'''N'''}}, and {{math|'''B'''}} can all be given as successive derivatives of the parametrization of the curve, each of them is insensitive to the addition of a constant vector to {{math|'''r'''(''t'')}}. Intuitively, the {{math|'''TNB'''}} frame attached to {{math|'''r'''(''t'')}} is the same as the {{math|'''TNB'''}} frame attached to the new curve {{math|'''r'''(''t'') + '''v'''}}.

This leaves only the rotations to consider. Intuitively, if we apply a rotation {{mvar|M}} to the curve, then the {{math|'''TNB'''}} frame also rotates. More precisely, the matrix {{mvar|Q}} whose rows are the {{math|'''TNB'''}} vectors of the Frenet–Serret frame changes by the matrix of a rotation

<math display=block> Q \rightarrow QM.</math>

''A fortiori'', the matrix <math>\tfrac{dQ}{ds}Q^\mathrm{T}</math> is unaffected by a rotation:

<math display=block>\frac{ \mathrm{d} (QM) }{ \mathrm{d} s} (QM)^\top = \frac{ \mathrm{d} Q}{ \mathrm{d} s } MM^\top Q^\top = \frac{ \mathrm{d} Q}{ \mathrm{d} s} Q^\top</math>

since {{math|1=''MM''<sup>T</sup> = ''I''}} for the matrix of a rotation.

Hence the entries {{mvar|κ}} and {{mvar|τ}} of <math>\tfrac{dQ}{ds}Q^\mathrm{T}</math> are ''invariants'' of the curve under Euclidean motions: if a Euclidean motion is applied to a curve, then the resulting curve has ''the same'' curvature and torsion.

Moreover, using the Frenet–Serret frame, one can also prove the converse: any two curves having the same curvature and torsion functions must be congruent by a Euclidean motion. Roughly speaking, the Frenet–Serret formulas express the [[Darboux derivative]] of the {{math|'''TNB'''}} frame. If the Darboux derivatives of two frames are equal, then a version of the [[fundamental theorem of calculus]] asserts that the curves are congruent. In particular, the curvature and torsion are a ''complete'' set of invariants for a curve in three-dimensions.