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Rectifying plane

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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