One can obtain two principal directions of a point (non-umbilic point) through solving a quadratic equation, i.e., $(LF-ME)(du)^2+(LG-NE)dudv+(MG-NF)(dv)^2=0$, but it is not easy to determine the information about which curvature line corresponds to which of the two principal directions from the equation above. Is there any way to tell that the principal directions derived at two different points both correspond to tangent vectors of curvature lines of the same family?
To be specific, as for the ellipsoid and an arbitrary regular surface generated from NURBS, how can I make the tangent vectors corresponding to two principal directions vary smoothly and continuously among different points on the surface, i.e., the two principal directions always correspond to the same family of curvature lines. (The principal directions of the same color in my case clearly do not constitute curvature lines of the same family)
Here I first calculate the principal directions and then the corresponding principal curvatures. Let $A = LF-ME, B = LG-NE, C = MG-NF$. Two solutions to the quadratic equation: $du:dv = \frac{-B\pm\sqrt{B^2-4AC}}{2A}$, thus the principal directions can be obtained by $d\mathbf{r} = \mathbf{r}_udu+\mathbf{r}_vdv$, and the principal curvatures are subsequently calculated by $k = \frac{Ldu+Mdv}{Edu+Fdv}$. Basis vectors at a point corresponding to two principal directions are determined just through normalization of $d\mathbf{r}$.
Basis vectors corresponding to two principal directions at several points on an ellipsoid Basis vectors corresponding to two principal directions at several points on an arbitrary surface generated from NURBS
