VectorCalculus

 compute the radius of curvature of a curve

Parameters

 C - free or position Vector or Vector valued procedure; specify the components of the curve t - (optional) name; specify the parameter of the curve

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$
 > $\mathrm{RadiusOfCurvature}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)⟩,t\right)$
 ${1}$ (1)
 > $\mathrm{RadiusOfCurvature}\left(\mathrm{PositionVector}\left(\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t\right]\right)\right)$
 ${2}$ (2)
 > $r≔\mathrm{RadiusOfCurvature}\left(t↦⟨t,{t}^{2},{t}^{3}⟩\right)$
 ${r}{≔}{t}{↦}\frac{\sqrt{{9}{\cdot }{{t}}^{{4}}{+}{4}{\cdot }{{t}}^{{2}}{+}{1}}}{{2}{\cdot }\sqrt{\frac{{9}{\cdot }{{t}}^{{4}}{+}{9}{\cdot }{{t}}^{{2}}{+}{1}}{{\left({9}{\cdot }{{t}}^{{4}}{+}{4}{\cdot }{{t}}^{{2}}{+}{1}\right)}^{{2}}}}}$ (3)
 > $\mathrm{RadiusOfCurvature}\left(⟨a\mathrm{cos}\left(t\right),a\mathrm{sin}\left(t\right),t⟩\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a::\mathrm{constant}$
 $\frac{\sqrt{{{a}}^{{2}}{+}{1}}}{\sqrt{\frac{{{a}}^{{2}}}{{{a}}^{{2}}{+}{1}}}}$ (4)
 > $\mathrm{SetCoordinates}\left('\mathrm{polar}'\right)$
 ${\mathrm{polar}}$ (5)
 > $\mathrm{RadiusOfCurvature}\left(⟨\mathrm{exp}\left(-t\right),t⟩\right)$
 $\sqrt{{2}}{}\sqrt{{{ⅇ}}^{{-}{2}{}{t}}}$ (6)