Given a point on a 2 dimensional surface embed in 3D, where exactly does a principle curve differ from a line of curvature? If we look at the definitions, it's not clear to me.
Principle curve: The curve that has $\alpha '(t)$ in the direction of the principle curvature.
Principle curvature: The maximum normal curvature at a given point on a surface.
Line of curvature: Curve that is always tangent to a principle curve
I don't see how that separates it from principle curves. The last definition would mean that $\alpha ' (t)$ is pointing along a principle direction, which would also make it a principle curve. What's the difference?