Student[VectorCalculus] - Maple Programming Help

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Student[VectorCalculus]

 compute the radius of curvature of a curve

Parameters

 C - free or position Vector; specify the components of the curve t - (optional) name; specify the parameter of the curve options - (optional) equation(s) of the form option=value where option is one of output, circleoptions, circles, curveoptions, range, or view

Description

 • The RadiusOfCurvature(C, t) calling sequence 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.
 • If t is not specified, the command 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 that 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.
 • The options arguments primarily control plot options.
 output = value, plot, or animation
 This option controls the return value of the command.
 – output = value returns the value of the radius of curvature. Plot options are ignored if output = value.  This is the default value.
 – output = plot returns a plot of the space curve and the circles. The number of circles is specified by the circles option. The RadiusOfCurvature command supports only three-dimensional Vector plots.
 – output = animation returns an animation of the space curve and the circles. The number of circles of curvature is specified by the circles option.
 • circleoptions = list
 A list of plot options for plotting the circles. For more information on plotting options, see plot/options. The default value is [].
 • circles = posint
 Specifies how many circles are to be plotted or animated. The default value is 5.
 • curveoptions = list
 A list of plot options for plotting the space curve. For more information on plotting options, see plot/options. The default value is [].
 • range = realcons..realcons
 The range of the independent variable. The default value is 0..5.
 • view = [realcons..realcons, realcons..realcons, realcons..realcons]
 • caption = anything
 A caption for the plot.
 The default caption is constructed from the parameters and the command options. caption = "" disables the default caption. For more information about specifying a caption, see plot/typesetting.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right):$
 > $\mathrm{RadiusOfCurvature}\left(\mathrm{PositionVector}\left(\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t\right]\right)\right)$
 ${2}$ (1)
 > $\mathrm{RadiusOfCurvature}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩\right)$
 ${2}$ (2)
 > $\mathrm{simplify}\left(\mathrm{Curvature}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩\right)\right)$
 $\frac{{1}}{{2}}$ (3)

To play the following animations in this help page, right-click (Control-click, on Macintosh) the plot to display the context menu.  Select Animation > Play.

 > $\mathrm{RadiusOfCurvature}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩,\mathrm{output}=\mathrm{animation},\mathrm{scaling}=\mathrm{constrained}\right)$
 > $\mathrm{RadiusOfCurvature}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩,\mathrm{output}=\mathrm{animation},\mathrm{circles}=20,\mathrm{scaling}=\mathrm{constrained},\mathrm{range}=0..10\right)$

The command to create the plot from the Plotting Guide is

 > $\mathrm{RadiusOfCurvature}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩,\mathrm{output}=\mathrm{plot},\mathrm{circles}=3,\mathrm{range}=0..10,\mathrm{scaling}=\mathrm{constrained},\mathrm{curveoptions}=\left[\mathrm{orientation}=\left[60,270\right]\right]\right)$