PrincipalNormal - Maple Help

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Student[VectorCalculus]

 PrincipalNormal
 compute a Vector in the direction of the normal vector to a curve in R^2 or R^3

 Calling Sequence PrincipalNormal(C, t, options)

Parameters

 C - free or position Vector; specify the components of the curve in R^2 or 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, curveoptions, normalized, range, vectoroptions, vectors, or view

Description

 • The PrincipalNormal(C, t) calling sequence computes a Vector in the direction of the normal vector to a curve in R^2 or R^3.
 Note: This vector is not normalized by default, so it is a scalar multiple of the unit normal vector to the curve C. Therefore, by default, the result is generally different from the output of TNBFrame(C, t, normal=true). If either normalized=true or normalized is given in options, however, the resulting vector will be normalized.
 • The curve C must have two or three components, that is, the curve that the Vector represents must be in R^2 or R^3.
 • 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, that 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 normal. Plot options are ignored if output = value.  This is the default value.
 – output = plot returns a plot of the space curve and the normal vectors. The number of normal vectors is specified by the vectors option.
 – output = animation returns an animation of the space curve and the normal vectors. The number of normal vectors is specified by the vectors option.
 • 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 [].
 • normalized = truefalse
 Either true or false. Specifies whether the normal vector is to be normalized.
 • range = realcons..realcons
 The range of the independent variable. The default value is 0..5.
 • vectoroptions = list
 A list of plot options for plotting the normal vectors. For more information on plotting options, see plot/options. The default value is []. Note: Free Vectors and rooted Vectors are plotted using plots[arrow].
 • vectors = posint
 Specifies how many normal vectors are to be plotted or animated. The default value is 5.
 • view = [realcons..realcons, realcons..realcons, realcons..realcons] (3-D) or [realcons..realcons, realcons..realcons] (2-D)
 Specify the plot view. For more information, see plot3d/options or plot/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

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right):$
 > $\mathrm{PrincipalNormal}\left(\mathrm{PositionVector}\left(\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t\right]\right),t\right)$
 $\left[\begin{array}{c}{-}\frac{\sqrt{{2}}{}{\mathrm{cos}}{}\left({t}\right)}{{2}}\\ {-}\frac{\sqrt{{2}}{}{\mathrm{sin}}{}\left({t}\right)}{{2}}\\ {0}\end{array}\right]$ (1)
 > $\mathrm{PrincipalNormal}\left(\mathrm{PositionVector}\left(\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t\right]\right),t,\mathrm{normalized}\right)$
 $\left[\begin{array}{c}{-}{\mathrm{cos}}{}\left({t}\right)\\ {-}{\mathrm{sin}}{}\left({t}\right)\\ {0}\end{array}\right]$ (2)
 > $\mathrm{PrincipalNormal}\left(⟨\mathrm{exp}\left(-t\right)\mathrm{cos}\left(t\right),\mathrm{exp}\left(-t\right)\mathrm{sin}\left(t\right),t⟩\right)$
 $\left[\begin{array}{c}{-}\frac{{2}{}{{ⅇ}}^{{-}{t}}{}\left({{ⅇ}}^{{-}{2}{}{t}}{}{\mathrm{cos}}{}\left({t}\right){-}{{ⅇ}}^{{-}{2}{}{t}}{}{\mathrm{sin}}{}\left({t}\right){-}{\mathrm{sin}}{}\left({t}\right)\right)}{{\left({1}{+}{2}{}{{ⅇ}}^{{-}{2}{}{t}}\right)}^{{3}}{{2}}}}\\ {-}\frac{{2}{}{{ⅇ}}^{{-}{t}}{}\left({{ⅇ}}^{{-}{2}{}{t}}{}{\mathrm{cos}}{}\left({t}\right){+}{{ⅇ}}^{{-}{2}{}{t}}{}{\mathrm{sin}}{}\left({t}\right){+}{\mathrm{cos}}{}\left({t}\right)\right)}{{\left({1}{+}{2}{}{{ⅇ}}^{{-}{2}{}{t}}\right)}^{{3}}{{2}}}}\\ \frac{{2}{}{{ⅇ}}^{{-}{2}{}{t}}}{{\left({1}{+}{2}{}{{ⅇ}}^{{-}{2}{}{t}}\right)}^{{3}}{{2}}}}\end{array}\right]$ (3)
 > $\mathrm{PrincipalNormal}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩,t,\mathrm{output}=\mathrm{plot},\mathrm{scaling}=\mathrm{constrained}\right)$
 > $\mathrm{PrincipalNormal}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩,t,\mathrm{output}=\mathrm{plot},\mathrm{range}=0..7,\mathrm{vectors}=3,\mathrm{vectoroptions}=\left[\mathrm{color}=\mathrm{red}\right],\mathrm{curveoptions}=\left[\mathrm{color}=\mathrm{green}\right],\mathrm{scaling}=\mathrm{constrained}\right)$

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.

 > $\mathrm{PrincipalNormal}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩,t,\mathrm{output}=\mathrm{animation},\mathrm{range}=0..7,\mathrm{vectors}=3,\mathrm{vectoroptions}=\left[\mathrm{color}=\mathrm{red}\right],\mathrm{curveoptions}=\left[\mathrm{color}=\mathrm{green}\right],\mathrm{scaling}=\mathrm{constrained}\right)$
 > $\mathrm{SetCoordinates}\left(\mathrm{cylindrical}\left[r,t,z\right]\right)$
 ${{\mathrm{cylindrical}}}_{{r}{,}{t}{,}{z}}$ (4)
 > $\mathrm{PrincipalNormal}\left(⟨a,t,t⟩\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a::\left(\mathrm{And}\left(\mathrm{positive},\mathrm{constant}\right)\right)$
 $\left[\begin{array}{c}{-}\frac{{a}}{\sqrt{{{a}}^{{2}}{+}{1}}}\\ {0}\\ {0}\end{array}\right]$ (5)
 > $\mathrm{SetCoordinates}\left(\mathrm{cartesian}\right)$
 ${\mathrm{cartesian}}$ (6)

The command to create the plot from the Plotting Guide is

 > $\mathrm{PrincipalNormal}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩,\mathrm{output}=\mathrm{plot},\mathrm{scaling}=\mathrm{constrained}\right)$