# Conjugate directions

From Encyclopedia of Mathematics

A pair of directions emanating from a point on a surface such that the straight lines containing them are conjugate diameters of the Dupin indicatrix of at . In order that the directions , at a point on be conjugate, it is necessary and sufficient that the following condition holds

where , and are the coefficients of the second fundamental form of evaluated at . 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. E.V. Shikin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Conjugate_directions&oldid=11253

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098