Principal Section

Differential equation of principal section/direction

Let S: \vec{r}=\vec{r}(u,v) be a surface and \kappa_n be the principal curvature, then
\kappa_n=\frac{Ldu^2+2Mdudv+Ndv^2}{Edu^2+2Fdudv+Gdv^2}
or (Ldu^2+2Mdudv+Ndv^2)-\kappa_n(Edu^2+2Fdudv+Gdv^2)=0
Then, differentiating w r. to. duanddv separately, we get
(Ldu+Mdv)-\kappa_n(Edu+Fdv)=0 and (Mdu+Ndv)-\kappa_n(Fdu+Gdv)=0
Eliminating k_n from (i) and (ii) we get
\frac{(Ldu+Mdv)}{(Mdu+Ndv)}=\frac{(Edu+Fdv)}{(Fdu+Gdv)}
or (EM-FL)du^2+(EN-GL)dudv+(FN-GM)dv^2=0 (i)
which is the required equation of principal directions.

Note
The equation of principal directions is
\begin{vmatrix}E&F&G\\L&M&N\\dv^2&-dudv&du^2\\\end{vmatrix}=0

Example

Find principal sections on hyperboloid 2z=7x^2+6xy-y^2 at origin
Solution

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