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The plane of the Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]]) of a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080140/r0801401.png" /> on a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080140/r0801402.png" /> (cf. [[Line (curve)|Line (curve)]]) which is spanned by the tangent (cf. [[Tangent line|Tangent line]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080140/r0801403.png" /> and the [[Binormal|binormal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080140/r0801404.png" /> to the curve at this point. The equation of the rectifying plane can be written in the form
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$#C+1 = 7 : ~/encyclopedia/old_files/data/R080/R.0800140 Rectifying plane
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080140/r0801405.png" /></td> </tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
or
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The plane of the Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]]) of a given point  $  A $
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on a curve  $  \mathbf r = \mathbf r ( t) $(
 +
cf. [[Line (curve)|Line (curve)]]) which is spanned by the tangent (cf. [[Tangent line|Tangent line]])  $  \mathbf t $
 +
and the [[Binormal|binormal]]  $  \mathbf b $
 +
to the curve at this point. The equation of the rectifying plane can be written in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080140/r0801406.png" /></td> </tr></table>
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$$
 +
\def\p{\prime}\def\pp{ {\p\p} }
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\left|
 +
\begin{matrix}
 +
X - x(A) & Y - y(A) & Z - z(A) \\
 +
  x^\p(A) &  y^\p(A) &  z^\p(A) \\
 +
\left|
 +
\begin{matrix}
 +
y^\p  & z^\p \\
 +
y^\pp & z^\pp\\
 +
\end{matrix}
 +
\right| &
 +
\left|
 +
\begin{matrix}
 +
z^\p  & x^\p \\
 +
z^\pp & x^\pp\\
 +
\end{matrix}
 +
\right| &
 +
\left|
 +
\begin{matrix}
 +
x^\p  & y^\p \\
 +
x^\pp & y^\pp\\
 +
\end{matrix}
 +
\right|
 +
\end{matrix}
 +
\right|
 +
= 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080140/r0801407.png" /> is the equation of the curve.
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or
  
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$$
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( \mathbf R - \mathbf r ) \mathbf r  ^  \prime  [ \mathbf r  ^  \prime  , \mathbf r
 +
^ {\prime\prime} ]  =  0,
 +
$$
  
 +
where  $  \mathbf r ( t) = \mathbf r ( x( t), y( t), z( t)) $
 +
is the equation of the curve.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''2''' , Publish or Perish  (1970)  pp. 1–5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''2''' , Publish or Perish  (1970)  pp. 1–5</TD></TR></table>

Latest revision as of 21:22, 7 June 2020


The plane of the Frénet frame (cf. Frénet trihedron) of a given point $ A $ on a curve $ \mathbf r = \mathbf r ( t) $( cf. Line (curve)) which is spanned by the tangent (cf. Tangent line) $ \mathbf t $ and the binormal $ \mathbf b $ to the curve at this point. The equation of the rectifying plane can be written in the form

$$ \def\p{\prime}\def\pp{ {\p\p} } \left| \begin{matrix} X - x(A) & Y - y(A) & Z - z(A) \\ x^\p(A) & y^\p(A) & z^\p(A) \\ \left| \begin{matrix} y^\p & z^\p \\ y^\pp & z^\pp\\ \end{matrix} \right| & \left| \begin{matrix} z^\p & x^\p \\ z^\pp & x^\pp\\ \end{matrix} \right| & \left| \begin{matrix} x^\p & y^\p \\ x^\pp & y^\pp\\ \end{matrix} \right| \end{matrix} \right| = 0, $$

or

$$ ( \mathbf R - \mathbf r ) \mathbf r ^ \prime [ \mathbf r ^ \prime , \mathbf r ^ {\prime\prime} ] = 0, $$

where $ \mathbf r ( t) = \mathbf r ( x( t), y( t), z( t)) $ is the equation of the curve.

Comments

References

[a1] M. Spivak, "A comprehensive introduction to differential geometry" , 2 , Publish or Perish (1970) pp. 1–5
How to Cite This Entry:
Rectifying plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rectifying_plane&oldid=49393
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article