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.

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

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

$\left[\begin{array}{c}-\frac{\sqrt{2}\cos \left(t\right)}{2}\\ -\frac{\sqrt{2}\sin \left(t\right)}{2}\\ 0\end{array}\right]$

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

$\left[\begin{array}{c}-\cos \left(t\right)\\ -\sin \left(t\right)\\ 0\end{array}\right]$

$\mathrm{PrincipalNormal}\left(⟨\exp \left(-t\right)\cos \left(t\right)\,\exp \left(-t\right)\sin \left(t\right)\,t⟩\right)$

$\left[\begin{array}{c}\frac{2\left({\ⅇ}^{-2t}\sin \left(t\right)-{\ⅇ}^{-2t}\cos \left(t\right)+\sin \left(t\right)\right){\ⅇ}^{-t}}{{\left(1+2{\ⅇ}^{-2t}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\\ -\frac{2{\ⅇ}^{-t}\left({\ⅇ}^{-2t}\sin \left(t\right)+{\ⅇ}^{-2t}\cos \left(t\right)+\cos \left(t\right)\right)}{{\left(1+2{\ⅇ}^{-2t}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\\ \frac{2{\ⅇ}^{-2t}}{{\left(1+2{\ⅇ}^{-2t}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\end{array}\right]$

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

$\mathrm{PrincipalNormal}\left(⟨\cos \left(t\right)\,\sin \left(t\right)\,t⟩\,t\,\mathrm{output}=\mathrm{plot}\,\mathrm{range}=0..7\,\mathrm{vectors}=3\,\mathrm{vectoroptions}=\left[\mathrm{color}=\mathrm{red}\right]\,\mathrm{curveoptions}=\left[\mathrm{color}=\mathrm{green}\right]\,\mathrm{scaling}=\mathrm{constrained}\right)$

$\mathrm{PrincipalNormal}\left(⟨\cos \left(t\right)\,\sin \left(t\right)\,t⟩\,t\,\mathrm{output}=\mathrm{animation}\,\mathrm{range}=0..7\,\mathrm{vectors}=3\,\mathrm{vectoroptions}=\left[\mathrm{color}=\mathrm{red}\right]\,\mathrm{curveoptions}=\left[\mathrm{color}=\mathrm{green}\right]\,\mathrm{scaling}=\mathrm{constrained}\right)$

$\mathrm{SetCoordinates}\left(\mathrm{cylindrical}\left[r,t,z\right]\right)$

${\mathrm{cylindrical}}_{r,t,z}$

$\mathrm{PrincipalNormal}\left(⟨a\,t\,t⟩\right)\mathbf{assuming}a::\left(\mathrm{And}\left(\mathrm{positive}\,\mathrm{constant}\right)\right)$

$\left[\begin{array}{c}-\frac{a}{\sqrt{a^2+1}}\\ 0\\ 0\end{array}\right]$

$\mathrm{SetCoordinates}\left(\mathrm{cartesian}\right)$

$\mathrm{cartesian}$

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