# 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[1]:= XSimplify[ArcCurvature[{r Cos[t], -r Sin[t]}, t], r > 0]
 Out[1]=

Curvature for a curve expressed in polar coordinates.

 In[2]:= XSimplify[ArcCurvature[{a t^2, t}, t, "Polar"], a > 0 && t > 0]
 Out[2]=

Curves in three and four dimensions.

 In[3]:= XSimplify[ArcCurvature[{r Cos[t], -r Sin[t], t}, t], r > 0]
 Out[3]=
 In[4]:= XSimplify[ArcCurvature[{r Cos[t], -r Sin[t], Cos[t], Sin[t]}, t], r > 0]
 Out[4]=

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

 In[5]:= XSimplify[FrenetSerretSystem[{r Cos[t], -r Sin[t]}, t], r > 0]
 Out[5]=

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

 In[6]:= XSimplify[FrenetSerretSystem[{r Cos[t], -r Sin[t], t}, t], r > 0]
 Out[6]=

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

 Out[7]=

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