# FrenetSerretSystem

FrenetSerretSystem[{x1,,xn},t]

gives the generalized curvatures and FrenetSerret basis for the parametric curve xi[t].

FrenetSerretSystem[{x1,,xn},t,chart]

interprets the xi as coordinates in the specified coordinate chart.

# Details • FrenetSerretSystem returns {{k1,,kn-1},{e1,,en}}, where ki are generalized curvatures and ei are the FrenetSerret basis vectors.
• The first basis vector e1 is the unit tangent to the curve. Each successive vector is the orthonormalized derivative of the previous one. The last vector is chosen to complete a right-handed orthonormal basis.
• • If one of the vector derivatives is zero, then the remaining vectors are also taken to be zero.
• Common names in dimension two and three are:
•  {{k1},{e1,e2}} signed curvature, tangent, and normal {{k1,k2},{e1,e2,e3}} curvature, torsion, tangent, normal, and binormal
• In FrenetSerretSystem[x,t], if x is a scalar expression, FrenetSerretSystem gives the curvature of the parametric curve {t,x}.
• If a chart is specified, the basis vectors ei are expressed in the orthonormal basis associated to it.
• Coordinate charts in the third argument of FrenetSerretSystem can be specified as triples {coordsys,metric,dim} in the same way as in the first argument of CoordinateChartData. The short form in which dim is omitted may be used.

# Examples

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## Basic Examples(2)

The curvature, tangent, and normal for a circle in two dimensions:

 In:= Out= The curvature, torsion, and associated basis for a helix expressed in cylindrical coordinates:

 In:= Out= In:= Out= ## Neat Examples(1)

Introduced in 2014
(10.0)