Rectifying plane

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.

References

[a1]	M. Spivak, "A comprehensive introduction to differential geometry" , 2 , Publish or Perish (1970) pp. 1–5