Difference between revisions of "Principal normal"
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| + | $#C+1 = 9 : ~/encyclopedia/old_files/data/P074/P.0704750 Principal normal | ||
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| + | A [[Normal|normal]] to a curve $ L $ | ||
| + | passing through a point $ M _ {0} $ | ||
| + | of $ L $ | ||
| + | and lying in the [[Osculating plane|osculating plane]] to $ L $ | ||
| + | at $ M _ {0} $. | ||
| + | If $ \mathbf r = \mathbf r ( t) $ | ||
| + | is the parametric equation of the curve and the value $ t _ {0} $ | ||
| + | corresponds to $ M _ {0} $, | ||
| + | then the equation of the principal normal in vector form is: | ||
| + | $$ | ||
| + | \mathbf r = \mathbf r ( t _ {0} ) + \lambda \mathbf r ^ {\prime\prime} ( t _ {0} ). | ||
| + | $$ | ||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.J. Struik, "Lectures in classical differential calculus" , Dover, reprint (1988) pp. 13</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 26</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.J. Struik, "Lectures in classical differential calculus" , Dover, reprint (1988) pp. 13</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 26</TD></TR></table> | ||
Latest revision as of 08:07, 6 June 2020
A normal to a curve $ L $
passing through a point $ M _ {0} $
of $ L $
and lying in the osculating plane to $ L $
at $ M _ {0} $.
If $ \mathbf r = \mathbf r ( t) $
is the parametric equation of the curve and the value $ t _ {0} $
corresponds to $ M _ {0} $,
then the equation of the principal normal in vector form is:
$$ \mathbf r = \mathbf r ( t _ {0} ) + \lambda \mathbf r ^ {\prime\prime} ( t _ {0} ). $$
Comments
References
| [a1] | D.J. Struik, "Lectures in classical differential calculus" , Dover, reprint (1988) pp. 13 |
| [a2] | R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 26 |
How to Cite This Entry:
Principal normal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_normal&oldid=48291
Principal normal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_normal&oldid=48291
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article