Editing Frenet–Serret formulas (section)

=== 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>