Difference between revisions of "Rectifying plane"
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| + | $#A+1 = 7 n = 0 | ||
| + | $#C+1 = 7 : ~/encyclopedia/old_files/data/R080/R.0800140 Rectifying plane | ||
| + | Automatically converted into TeX, above some diagnostics. | ||
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| + | if TeX found to be correct. | ||
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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 | ||
| − | + | $$ | |
| + | \left | | ||
| + | \begin{array}{cllcllcll} | ||
| + | X - x( A) &\left | | ||
| + | \begin{array}{} | ||
| + | y ^ \prime &z ^ \prime &Y - y( A) &\left | | ||
| + | \begin{array}{} | ||
| + | z ^ \prime &x ^ \prime &Z - z( A) &\left | | ||
| + | \begin{array}{} | ||
| + | x ^ \prime &y ^ \prime \\ | ||
| + | x ^ \prime ( A) &y ^ {\prime\prime} &z ^ {\prime\prime} \\ | ||
| + | \end{array} | ||
| + | \\ | ||
| + | \end{array} | ||
| + | \\ | ||
| + | \end{array} | ||
| + | \right | &y ^ \prime ( A) &z ^ {\prime\prime} &x ^ {\prime\prime} \right | &z ^ \prime ( A) &x ^ {\prime\prime} &y ^ {\prime\prime} \right | \\ | ||
| + | \end{array} | ||
| + | \right | | ||
| + | = 0, | ||
| + | $$ | ||
| − | + | 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==== | ====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 14:55, 7 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
$$ \left | \begin{array}{cllcllcll} X - x( A) &\left | \begin{array}{} y ^ \prime &z ^ \prime &Y - y( A) &\left | \begin{array}{} z ^ \prime &x ^ \prime &Z - z( A) &\left | \begin{array}{} x ^ \prime &y ^ \prime \\ x ^ \prime ( A) &y ^ {\prime\prime} &z ^ {\prime\prime} \\ \end{array} \\ \end{array} \\ \end{array} \right | &y ^ \prime ( A) &z ^ {\prime\prime} &x ^ {\prime\prime} \right | &z ^ \prime ( A) &x ^ {\prime\prime} &y ^ {\prime\prime} \right | \\ \end{array} \right | = 0, $$
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 |
Rectifying plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rectifying_plane&oldid=49393