# Compute the Curvature of Curves in Any Dimensions

ArcCurvature and FrenetSerretSystem compute curvatures for curves in any dimension.

ArcCurvature gives the single unsigned curvature.

 In:= XSimplify[ArcCurvature[{r Cos[t], -r Sin[t]}, t], r > 0]
 Out= Curvature for a curve expressed in polar coordinates.

 In:= XSimplify[ArcCurvature[{a t^2, t}, t, "Polar"], a > 0 && t > 0]
 Out= Curves in three and four dimensions.

 In:= XSimplify[ArcCurvature[{r Cos[t], -r Sin[t], t}, t], r > 0]
 Out= In:= XSimplify[ArcCurvature[{r Cos[t], -r Sin[t], Cos[t], Sin[t]}, t], r > 0]
 Out= FrenetSerretSystem gives the generalized curvatures, which may be signed, and the associated basis.

 In:= XSimplify[FrenetSerretSystem[{r Cos[t], -r Sin[t]}, t], r > 0]
 Out= In three dimensions, the generalized curvatures are usually called curvature and torsion, and the associated Tangent/Normal/Binormal or TNB basis.

 In:= XSimplify[FrenetSerretSystem[{r Cos[t], -r Sin[t], t}, t], r > 0]
 Out= Visualize the four curves. The fourth dimension is represented by color.

 Out= ## Mathematica

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