PrincipalNormal - Maple Help

VectorCalculus

 PrincipalNormal
 compute a Vector in the direction of the principal normal vector to a curve

 Calling Sequence PrincipalNormal(C, t, n)

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 n - (optional) equation of the form normalized=true or normalized=false, or simply normalized

Description

 • The PrincipalNormal(C, t) command computes a Vector in the direction of the principal normal vector to the curve C.  Note that 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, if C is a curve in R^3, the result is generally different from the output of TNBFrame(C, t, output=['N']).
 • If n is given as either normalized=true or normalized, then the resulting vector will be normalized before it is returned. As discussed above, the default value is false, so that the result is not normalized.
 • The curve C can be specified as a free or position Vector or a Vector valued procedure. This determines the returned object type.
 • If t is not specified, the function 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 which are determined to be constants.
 If the resulting set has a single entry, that 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 that 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

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$
 > $\mathrm{PrincipalNormal}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)⟩,t\right)$
 $\left[\begin{array}{c}{-}{\mathrm{cos}}{}\left({t}\right)\\ {-}{\mathrm{sin}}{}\left({t}\right)\end{array}\right]$ (1)
 > $\mathrm{PrincipalNormal}\left(\mathrm{PositionVector}\left(\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t\right]\right)\right)$
 $\left[\begin{array}{c}{-}\frac{\sqrt{{2}}{}{\mathrm{cos}}{}\left({t}\right)}{{2}}\\ {-}\frac{\sqrt{{2}}{}{\mathrm{sin}}{}\left({t}\right)}{{2}}\\ {0}\end{array}\right]$ (2)
 > $\mathrm{P1}≔\mathrm{PrincipalNormal}\left(t↦⟨t,{t}^{2},{t}^{3}⟩\right):$
 > $\mathrm{P1}\left(t\right)$
 $\left[\begin{array}{c}{-}\frac{{2}{}{t}{}\left({9}{}{{t}}^{{2}}{+}{2}\right)}{{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\\ {-}\frac{{2}{}\left({9}{}{{t}}^{{4}}{-}{1}\right)}{{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\\ \frac{{6}{}{t}{}\left({2}{}{{t}}^{{2}}{+}{1}\right)}{{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\end{array}\right]$ (3)
 > $\mathrm{P2}≔\mathrm{PrincipalNormal}\left(t↦⟨t,{t}^{2},{t}^{3}⟩,\mathrm{normalized}\right):$
 > $\mathrm{P2}\left(t\right)$
 $\left[\begin{array}{c}{-}\frac{{t}{}\left({9}{}{{t}}^{{2}}{+}{2}\right)}{\sqrt{\frac{{9}{}{{t}}^{{4}}{+}{9}{}{{t}}^{{2}}{+}{1}}{{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{2}}}}{}{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\\ {-}\frac{{9}{}{{t}}^{{4}}{-}{1}}{\sqrt{\frac{{9}{}{{t}}^{{4}}{+}{9}{}{{t}}^{{2}}{+}{1}}{{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{2}}}}{}{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\\ \frac{{3}{}{t}{}\left({2}{}{{t}}^{{2}}{+}{1}\right)}{\sqrt{\frac{{9}{}{{t}}^{{4}}{+}{9}{}{{t}}^{{2}}{+}{1}}{{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{2}}}}{}{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\end{array}\right]$ (4)
 > $\mathrm{SetCoordinates}\left('\mathrm{polar}'\right)$
 ${\mathrm{polar}}$ (5)
 > $\mathrm{PrincipalNormal}\left(⟨a\mathrm{exp}\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},0
 $\left[\begin{array}{c}{-}\frac{\sqrt{{2}}}{{2}}\\ {-}\frac{\sqrt{{2}}}{{2}}\end{array}\right]$ (6)