- Frenet–Serret formulas
:"Binormal" redirects here. For the category-theoretic meaning of this word, see

Normal morphism ."In

vector calculus , the**Frenet–Serret formulas**describe the kinematic properties of a particle which moves along a continuous, differentiablecurve in three-dimensionalEuclidean space **R**^{"3"}. More specifically, the formulas describe thederivative s of the so-called**tangent, normal, and binormal**unit vector s in terms of each other. The formulas are named after the two French mathematicians who independently discovered them:Jean Frédéric Frenet , in his thesis of 1847, andJoseph Alfred Serret in 1851. Vector notation and linear algebra currently used to write these formulas was not yet in use at the time of their discovery.The tangent, normal, and binormal vectors, often called

**T**,**N**, and**B**, or collectively the**Frenet–Serret frame**or**TNB frame**are defined as follows:

***T**is the unit vector tangent to the curve, pointing in the direction of motion.

***N**is the derivative of**T**with respect to the arclength parameter of the curve, divided by its length.

***B**is thecross product of "T" and "N".The Frenet–Serret formulas are:$egin\{matrix\}frac\{dmathbf\{T\{ds\}\; =\; kappa\; mathbf\{N\}\; \backslash \backslash frac\{dmathbf\{N\{ds\}\; =\; -\; kappa\; mathbf\{T\}\; +,\; au\; mathbf\{B\}\backslash \backslash frac\{dmathbf\{B\{ds\}\; =\; -\; au\; mathbf\{N\}\; end\{matrix\}$where "d"/"ds" is the derivative with respect to arclength, κ is thecurvature and τ is the torsion of the curve. This formula effectively defines the curvature and torsion of a space curve.**Frenet–Serret formulas**Let

**r**(t) be acurve inEuclidean space , representing theposition vector of the particle as a function of time. The Frenet–Serret formulas apply to curves which are "non-degenerate", which roughly means that they havecurvature . More formally, in this situation thevelocity vector**r**′(t) and theacceleration vector**r**′′(t) are required not to be proportional.

* Serret, J. A. [*http://math-doc.ujf-grenoble.fr/cgi-bin/jmpar.py?O=16395&E=00000507&N=2&CD=0&F=PDF "Sur quelques formules relatives à la théorie des courbes à double courbure."*] "J. de Math."**16**, 1851.

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* Struik, Dirk J., "Lectures on Classical Differential Geometry", Addison-Wesley, Reading, Mass, 1961.

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* Hanson, A.J. [*http://www.cs.indiana.edu/pub/techreports/TR407.pdf Quaternion Frenet Frames: Making Optimal Tubes and Ribbons from Curves*] , "Indiana University Technical Report."

* Goriely, A., Robertson-Tessi, M., Tabor, M., Vandiver, R. (2006) [*http://math.arizona.edu/~goriely/Papers/2006-biomat.pdf Elastic growth models*] , BIOMAT-2006, Springer-Verlag.

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***External links*** [

*http://www.mathcs.sjsu.edu/faculty/rucker/kaptaudoc/ktpaper.htm Rudy Rucker's KappaTau Paper*] .

* [*http://www.math.byu.edu/~math302/content/learningmod/trihedron/trihedron.html Very nice visual representation for the trihedron*]

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**Jean Frédéric Frenet**— (February 7, 1816 – June 12, 1900) was a French mathematician, astronomer, and meteorologist. He was born and died in Périgueux, France.He is best known for being an (independent) co discoverer of the Frenet Serret formulas. He wrote six out of… … Wikipedia**Joseph Alfred Serret**— (August 301819 March 2, 1885) was a French mathematician who was born in Paris France and died in Versailles France.ee also*Frenet Serret formulasBooks by J. A. Serret* [http://name.umdl.umich.edu/ABN8403.0001.001 Traité de trigonométrie]… … Wikipedia**Jean Frenet**— Nacimiento 7 de febrero de 1816 Francia, Périgueux Fallecimiento 12 de junio de … Wikipedia Español**Differential geometry of curves**— This article considers only curves in Euclidean space. Most of the notions presented here have analogues for curves in Riemannian and pseudo Riemannian manifolds. For a discussion of curves in an arbitrary topological space, see the main article… … Wikipedia**Darboux frame**— In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non umbilic point of a surface … Wikipedia**Moving frame**— The Frenet Serret frame on a curve is the simplest example of a moving frame. In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry… … Wikipedia**Torsion tensor**— In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet Serret formulas, for instance, quantifies the twist of a curve… … Wikipedia**Centripetal force**— Not to be confused with Centrifugal force. Classical mechanics Newton s Second Law … Wikipedia**Mechanics of planar particle motion**— Classical mechanics Newton s Second Law History of classical mechanics … Wikipedia**Curvature**— In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this … Wikipedia