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As much as i understand. Geodesic line of curvature is a line on the surface such that the projection on the tangent plane of it's curvature vector is 0 at every point. The lines of curavture are lines such that their tangent direction is coincident with one of the principal direction on the surface at every point. But I can't see the difference between them. Can anyone make it clear for me?

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One obvious difference: through a given point there are geodesics going in any direction, not only in the principal directions. –  Hans Lundmark Jun 20 '12 at 14:08
wow, that was clear. Thanks a lot. Stupid me –  Mykolas Jun 20 '12 at 14:20

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