# Conjugate directions

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) [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=53806
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article