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.

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