A curve on a surface whose tangents are always in the direction of principal curvature. The equation of the lines of curvature can be written left bracketing bar g_11 | g_12 | g_22 b_11 | b_12 | b_22 d u^2 | -d u d v | d v^2 right bracketing bar = 0, where g and b are the coefficients of the first and second fundamental forms.

Dupin's theorem | fundamental forms | principal curvatures