View source for Rectifying plane

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The plane of the Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]]) of a given point  $  A $
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

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

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====
<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>