ArcCurvature

ArcCurvature[{x1,,xn},t]
gives the curvature of the parametrized curve whose Cartesian coordinates are functions of t.

ArcCurvature[{x1,,xn},t,chart]
interprets the as coordinates in the specified coordinate chart.

DetailsDetails

  • The arc curvature is sometimes referred to as the unsigned or Frenet curvature.
  • The arc curvature of the curve in three-dimensional Euclidean space is given by (TemplateBox[{{{{x, ^, {(, ', )}}, (, t, )}, x, {{x, ^, {(, {', '}, )}}, , {(, t, )}}}}, Norm])/(TemplateBox[{{{x, ^, {(, ', )}}, (, t, )}}, Norm]^3).
  • In a general space, the arc curvature of the curve is given by TemplateBox[{{{D, /, {(, {D, , t}, )}}, {{(, { , {{x, ^, {(, ', )}}, (, t, )}}, )}, /, {(, TemplateBox[{{{x, ^, {(, ', )}}, (, t, )}}, Norm], )}}}}, Norm].
  • In ArcCurvature[x,t], if x is a scalar expression, ArcCurvature gives the curvature of the parametric curve .
  • Coordinate charts in the third argument of ArcCurvature can be specified as triples 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)

A circle has constant curvature:

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The curvature of Fermat's spiral expressed in polar coordinates:

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Visualize both branches of the curve:

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Introduced in 2014
(10.0)