Difference between revisions of "Rectifying plane"
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| + | $#C+1 = 7 : ~/encyclopedia/old_files/data/R080/R.0800140 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 | ||
| − | + | $$ | |
| + | \left | | ||
| − | + | or | |
| + | $$ | ||
| + | ( \mathbf R - \mathbf r ) \mathbf r ^ \prime [ \mathbf r ^ \prime , \mathbf r | ||
| + | ^ {\prime\prime} ] = 0, | ||
| + | $$ | ||
| + | where $ \mathbf r ( t) = \mathbf r ( x( t), y( t), z( t)) $ | ||
| + | is the equation of the curve. | ||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Spivak, "A comprehensive introduction to differential geometry" , '''2''' , Publish or Perish (1970) pp. 1–5</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Spivak, "A comprehensive introduction to differential geometry" , '''2''' , Publish or Perish (1970) pp. 1–5</TD></TR></table> | ||
Revision as of 08:10, 6 June 2020
The plane of the Frénet frame (cf. Frénet trihedron) of a given point $ A $
on a curve $ \mathbf r = \mathbf r ( t) $(
cf. Line (curve)) which is spanned by the tangent (cf. Tangent line) $ \mathbf t $
and the binormal $ \mathbf b $
to the curve at this point. The equation of the rectifying plane can be written in the form
$$ \left | 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
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