ArcCurvature

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

ArcCurvature[{x1,,xn},t,chart]
interprets the xi 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 {t,x}.
  • Coordinate charts in the third argument of ArcCurvature 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)

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)