Namespaces
Variants
Actions

Difference between revisions of "Rectifying plane"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (typos)
 
(2 intermediate revisions by the same user not shown)
Line 17: Line 17:
 
to the curve at this point. The equation of the rectifying plane can be written in the form
 
to the curve at this point. The equation of the rectifying plane can be written in the form
  
$$  
+
$$
\left |
+
\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
 
or

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=48452
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article