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


interprets the xi as coordinates in the specified coordinate chart.


  • 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.


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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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Scope  (6)

Applications  (3)

Properties & Relations  (7)

Interactive Examples  (1)

Neat Examples  (1)

See Also

ArcCurvature  ArcLength  CoordinateChartData  KnotData  D  Orthogonalize  CoordinateTransform

Introduced in 2014