The RadiusOfCurvature(C, t) command computes the radius of curvature of the curve C.  This is defined to be 1/Curvature(C, t) when the curvature is not zero and infinity when the curvature is zero.

The curve C can be specified as a free or positionVector or a Vector valued procedure, which determines the returned object type.

If t is not specified, the function tries to determine a suitable variable name from the components of C.  To do this, it checks all of the indeterminates of type name in the components of C and removes the ones which are determined to be constants.

If the resulting set has a single entry, this single entry is the variable name.  If it has more than one entry, an error is raised.

If a coordinate system attribute is specified on C, C is interpreted in this coordinate system.  Otherwise, the curve is interpreted as a curve in the current default coordinate system.  If the two are not compatible, an error is raised.

$\mathrm{with}\left(\mathrm{VectorCalculus}\right)\:$

$\mathrm{RadiusOfCurvature}\left(⟨\cos \left(t\right)\,\sin \left(t\right)⟩\,t\right)$

$1$

$\mathrm{RadiusOfCurvature}\left(\mathrm{PositionVector}\left(\left[\cos \left(t\right)\,\sin \left(t\right)\,t\right]\right)\right)$

$2$

$r≔\mathrm{RadiusOfCurvature}\left(t\mapsto ⟨t\,t^2\,t^3⟩\right)$

$r≔t\mapsto \frac{\sqrt{9\cdot t^4+4\cdot t^2+1}}{2\cdot \sqrt{\frac{9\cdot t^4+9\cdot t^2+1}{{\left(9\cdot t^4+4\cdot t^2+1\right)}^2}}}$

$\mathrm{RadiusOfCurvature}\left(⟨a\cos \left(t\right)\,a\sin \left(t\right)\,t⟩\right)\mathbf{assuming}a::\mathrm{constant}$

$\frac{\sqrt{a^2+1}}{\sqrt{\frac{a^2}{a^2+1}}}$

$\mathrm{SetCoordinates}\left('\mathrm{polar}'\right)$

$\mathrm{polar}$

$\mathrm{RadiusOfCurvature}\left(⟨\exp \left(-t\right)\,t⟩\right)$

$\sqrt{2}\sqrt{{\ⅇ}^{-2t}}$