TNBFrame - Maple Help

Student[VectorCalculus]

 TNBFrame
 compute the tangent-normal-binormal frame of a curve in R^3

 Calling Sequence TNBFrame(C, t, options)

Parameters

 C - free or position Vector; specify the components of the curve in R^3 t - (optional) name; specify the parameter of the curve options - (optional) equation(s) of the form option=value where option is one of output, binormal, binormaloptions, curveoptions, frames, normal, normaloptions, range, tangent, tangentoptions, or view

Description

 • The TNBFrame(C, t, out) calling sequence computes the tangent-normal-binormal frame of the curve C, that is, the unit tangent, normal, and binormal Vectors.  The computed objects are returned in an expression sequence.
 • If t is not specified, the command tries to determine a suitable variable name by using 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, the 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. The output is always returned in the order: tangent, normal, binormal.
 – output = value returns the value of the binormal. Plot options are ignored if output = value.  This is the default value.
 – output = plot returns a plot of the space curve and the TNB frames. The number of TNB frames is specified by the frames option.
 – output = animation returns an animation of the space curve and the TNB frames. The number of binormal frames is specified by the frames option.
 • binormal = truefalse
 Controls whether the binormal vector is included in the output or plot. The default value is true.
 • binormaloptions = list
 A list of plot options for plotting the binormal portion of the TNB frame. For more information on plotting options, see plot/options. The default value is []. Note: Vectors are plotted using plots[arrow].
 • 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 [].
 • frames = posint
 Specifies how many TNB frames are to be plotted or animated. The default value is 5.
 • normal = truefalse
 Controls whether the normal vector is included in the output or plot. The default value is true.
 • normaloptions = list
 A list of plot options for plotting the normal portion of the TNB frame. For more information on plotting options, see plot/options. The default value is []. Note: Vectors are plotted using plots[arrow].
 • range = realcons..realcons
 The range of the independent variable. The default value is 0..5.
 • tangent = truefalse
 Controls whether the tangent vector is included in the output or plot. The default value is true.
 • tangentoptions = list
 A list of plot options for plotting the tangent portion of the TNB frame. For more information on plotting options, see plot/options. The default value is []. Note: Vectors are plotted using plots[arrow].
 • view = [realcons..realcons, realcons..realcons, realcons..realcons]
 Specify the plot view. For more information, see plot3d/options.
 • 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

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right):$
 > $\mathrm{TNBFrame}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩,t\right)$
 $\left[\begin{array}{c}{-}\frac{\sqrt{{2}}{}{\mathrm{sin}}{}\left({t}\right)}{{2}}\\ \frac{\sqrt{{2}}{}{\mathrm{cos}}{}\left({t}\right)}{{2}}\\ \frac{\sqrt{{2}}}{{2}}\end{array}\right]{,}\left[\begin{array}{c}{-}{\mathrm{cos}}{}\left({t}\right)\\ {-}{\mathrm{sin}}{}\left({t}\right)\\ {0}\end{array}\right]{,}\left[\begin{array}{c}\frac{\sqrt{{2}}{}{\mathrm{sin}}{}\left({t}\right)}{{2}}\\ {-}\frac{\sqrt{{2}}{}{\mathrm{cos}}{}\left({t}\right)}{{2}}\\ \frac{\sqrt{{2}}}{{2}}\end{array}\right]$ (1)
 > $\mathrm{TNBFrame}\left(\mathrm{PositionVector}\left(\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)\right]\right)\right)$
 $\left[\begin{array}{c}{-}{\mathrm{sin}}{}\left({t}\right)\\ {\mathrm{cos}}{}\left({t}\right)\end{array}\right]{,}\left[\begin{array}{c}{-}{\mathrm{cos}}{}\left({t}\right)\\ {-}{\mathrm{sin}}{}\left({t}\right)\end{array}\right]$ (2)

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

 > $\mathrm{TNBFrame}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),\frac{t}{3}⟩,\mathrm{range}=0..2\mathrm{\pi },\mathrm{output}=\mathrm{animation},\mathrm{scaling}=\mathrm{constrained},\mathrm{axes}=\mathrm{frame},\mathrm{frames}=15\right)$
 > $\mathrm{SetCoordinates}\left(\mathrm{cylindrical}\left[r,t,z\right]\right)$
 ${{\mathrm{cylindrical}}}_{{r}{,}{t}{,}{z}}$ (3)
 > $\mathrm{TNBFrame}\left(⟨1,t,t⟩\right)$
 $\left[\begin{array}{c}\frac{{-}\frac{{\mathrm{cos}}{}\left({t}\right){}{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right){}\sqrt{{2}}{}{\mathrm{sin}}{}\left({t}\right)}{{2}}{+}\frac{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right){}\sqrt{{2}}}{{2}}}{\sqrt{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}{+}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}}}\\ \frac{\frac{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right){}\sqrt{{2}}}{{2}}{+}\frac{{\mathrm{cos}}{}\left({t}\right){}{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right){}\sqrt{{2}}{}{\mathrm{sin}}{}\left({t}\right)}{{2}}}{\sqrt{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}{+}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}}}\\ \frac{\sqrt{{2}}}{{2}}\end{array}\right]{,}\left[\begin{array}{c}\frac{{-}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right){-}{\mathrm{cos}}{}\left({t}\right){}{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right){}{\mathrm{sin}}{}\left({t}\right)}{\sqrt{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}{+}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}}}\\ \frac{{-}{\mathrm{cos}}{}\left({t}\right){}{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right){}{\mathrm{sin}}{}\left({t}\right){+}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}{\sqrt{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}{+}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}}}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}\frac{\frac{{\mathrm{cos}}{}\left({t}\right){}{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right){}\sqrt{{2}}{}{\mathrm{sin}}{}\left({t}\right)}{{2}}{-}\frac{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right){}\sqrt{{2}}}{{2}}}{\sqrt{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}{+}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}}}\\ \frac{{-}\frac{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right){}\sqrt{{2}}}{{2}}{-}\frac{{\mathrm{cos}}{}\left({t}\right){}{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right){}\sqrt{{2}}{}{\mathrm{sin}}{}\left({t}\right)}{{2}}}{\sqrt{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}{+}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{}{{\mathrm{cos}}{}\left({\mathrm{sin}}{}\left({t}\right)\right)}^{{2}}}}\\ \frac{\sqrt{{2}}}{{2}}\end{array}\right]$ (4)
 > $\mathrm{SetCoordinates}\left(\mathrm{cartesian}\right)$
 ${\mathrm{cartesian}}$ (5)

The command to create the plot from the Plotting Guide is

 > $\mathrm{TNBFrame}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩,\mathrm{output}=\mathrm{plot},\mathrm{binormal}=\mathrm{false},\mathrm{scaling}=\mathrm{constrained},\mathrm{normaloptions}=\left[\mathrm{orientation}=\left[240,130\right]\right]\right)$