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Difference between revisions of "Rectifying plane"

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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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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) $(
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cf. [[Line (curve)|Line (curve)]]) which is spanned by the tangent (cf. [[Tangent line|Tangent line]])  $  \mathbf t $
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and the [[Binormal|binormal]]  $  \mathbf b $
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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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$$
 +
\left |
 +
\begin{array}{cllcllcll}
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X - x( A)  &\left |
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\begin{array}{}
 +
y  ^  \prime  &z  ^  \prime  &Y - y( A)  &\left |
 +
\begin{array}{}
 +
z  ^  \prime  &x  ^  \prime  &Z - z( A)  &\left |
 +
\begin{array}{}
 +
x  ^  \prime  &y  ^  \prime  \\
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x  ^  \prime  ( A)  &y  ^ {\prime\prime}  &z  ^ {\prime\prime}  \\
 +
\end{array}
 +
  \\
 +
\end{array}
 +
  \\
 +
\end{array}
 +
  \right |  &y  ^  \prime  ( A)  &z  ^ {\prime\prime}  &x  ^ {\prime\prime}  \right |  &z  ^  \prime  ( A)  &x  ^ {\prime\prime}  &y  ^ {\prime\prime}  \right |  \\
 +
\end{array}
 +
\right |
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= 0,
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$$
  
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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$$
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( \mathbf R - \mathbf r ) \mathbf r  ^  \prime  [ \mathbf r  ^  \prime  , \mathbf r
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^ {\prime\prime} ]  =  0,
 +
$$
  
 +
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>

Revision as of 14:55, 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

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

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