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.

DetailsDetails

  • 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.

ExamplesExamplesopen allclose all

Basic Examples  (2)Basic Examples  (2)

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

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

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Introduced in 2014
(10.0)