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A pair of directions emanating from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c0250001.png" /> on a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c0250002.png" /> such that the straight lines containing them are conjugate diameters of the [[Dupin indicatrix|Dupin indicatrix]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c0250003.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c0250004.png" />. In order that the directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c0250005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c0250006.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c0250007.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c0250008.png" /> be conjugate, it is necessary and sufficient that the following condition holds
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c0250009.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c02500010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c02500011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c02500012.png" /> are the coefficients of the second fundamental form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c02500013.png" /> evaluated at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025000/c02500014.png" />. Example: a [[Principal direction|principal direction]].
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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>
 
 
  
 
====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
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article