# 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)