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

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

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.

References

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