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
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
or (EM-FL)du^2+(EN-GL)dudv+(FN-GM)dv^2=0 (i)
which is the required equation of principal directions.

The equation of principal directions is


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

Leave a Reply

Your email address will not be published. Required fields are marked *