Difference between revisions of "Conjugate directions"
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| − | + | A pair of directions emanating from a point $ P $ | |
| + | on a surface $ S $ | ||
| + | such that the straight lines containing them are conjugate diameters of the [[Dupin indicatrix|Dupin indicatrix]] of $ S $ | ||
| + | at $ P $. | ||
| + | In order that the directions $ ( du : dv) $, | ||
| + | $ ( \delta u : \delta v) $ | ||
| + | at a point $ P $ | ||
| + | on $ S $ | ||
| + | be conjugate, it is necessary and sufficient that the following condition holds | ||
| + | |||
| + | $$ | ||
| + | L du \delta u + M ( du \delta v + dv \delta u) + | ||
| + | N dv \delta v = 0, | ||
| + | $$ | ||
| + | |||
| + | where $ L $, | ||
| + | $ M $ | ||
| + | and $ N $ | ||
| + | are the coefficients of the second fundamental form of $ S $ | ||
| + | evaluated at $ P $. | ||
| + | Example: a [[Principal direction|principal direction]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , '''1''' , Springer (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , '''1''' , Springer (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4</TD></TR></table> | ||
Revision as of 17:46, 4 June 2020
A pair of directions emanating from a point $ P $
on a surface $ S $
such that the straight lines containing them are conjugate diameters of the Dupin indicatrix of $ S $
at $ P $.
In order that the directions $ ( du : dv) $,
$ ( \delta u : \delta v) $
at a point $ P $
on $ S $
be conjugate, it is necessary and sufficient that the following condition holds
$$ L du \delta u + M ( du \delta v + dv \delta u) + N dv \delta v = 0, $$
where $ L $, $ M $ and $ N $ are the coefficients of the second fundamental form of $ S $ evaluated at $ P $. Example: a principal direction.
References
| [1] | A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian) |
Comments
References
| [a1] | W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , 1 , Springer (1973) |
| [a2] | C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4 |
How to Cite This Entry:
Conjugate directions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_directions&oldid=46468
Conjugate directions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_directions&oldid=46468
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article