Compute the Curvature of Curves in Any Dimensions 

ArcCurvature and FrenetSerretSystem compute curvatures for curves in any dimension.

ArcCurvature gives the single unsigned curvature.

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X Simplify[ArcCurvature[{r Cos[t], -r Sin[t]}, t], r > 0]

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Curvature for a curve expressed in polar coordinates.

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X Simplify[ArcCurvature[{a t^2, t}, t, "Polar"], a > 0 && t > 0]

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Curves in three and four dimensions.

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X Simplify[ArcCurvature[{r Cos[t], -r Sin[t], t}, t], r > 0]

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X Simplify[ArcCurvature[{r Cos[t], -r Sin[t], Cos[t], Sin[t]}, t], r > 0]

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FrenetSerretSystem gives the generalized curvatures, which may be signed, and the associated basis.

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X Simplify[FrenetSerretSystem[{r Cos[t], -r Sin[t]}, t], r > 0]

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In three dimensions, the generalized curvatures are usually called curvature and torsion, and the associated Tangent/Normal/Binormal or TNB basis.

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X Simplify[FrenetSerretSystem[{r Cos[t], -r Sin[t], t}, t], r > 0]

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Visualize the four curves. The fourth dimension is represented by color.

show complete Wolfram Language inputhide input

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X Grid[{{ParametricPlot[{2 Cos[t], -2 Sin[t]}, {t, 0, 2 Pi}], PolarPlot[{2 Sqrt[t], - 2 Sqrt[t]}, {t, 0, 4 Pi}, PlotStyle -> ColorData[97, 1]]}, {ParametricPlot3D[{2 Cos[t], -2 Sin[t], t}, {t, 0, 4 Pi}, PlotTheme -> {"Default", "Scientific"}], ParametricPlot3D[{2 Cos[t], -2 Sin[t], Cos[t]}, {t, 0, 2 Pi}, ColorFunction -> (Hue[Sin[#4]] &), PlotTheme -> "Scientific"]}}, Dividers -> All]

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