# Conjugate directions

From Encyclopedia of Mathematics

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