Frenet–Serret formulas


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, differentiable curve in three-dimensional Euclidean space R"3". More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors 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, and Joseph 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 the cross product of "T" and "N".The Frenet–Serret formulas are: egin{matrix}frac{dmathbf{T{ds} &=& & kappa mathbf{N} & \&&&&\frac{dmathbf{N{ds} &=& - kappa mathbf{T} & &+, au mathbf{B}\&&&&\frac{dmathbf{B{ds} &=& & - au mathbf{N} &end{matrix}where "d"/"ds" is the derivative with respect to arclength, κ is the curvature 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 a curve in Euclidean space, representing the position 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 have curvature. More formally, in this situation the velocity vector r′(t) and the acceleration 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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