# Principal curvatures and directions of a polynomial surface

Given a polynomial surface $M =\{(x,y,z): z = p(x,y))\}$ where $p(x,y)$ is a polynomial. I want compute the principal curvatures and directions at some point in $M$. But I do not have idea to compute the principal directions.

For the part of principal curvatures, I can formulate $E、F、G、L、M、N、W$, which are coefficients in first and second fundamental form, in function of $p,p_x,p_{xx},p_y,p_{yy},p_{xy}$. Then the Mean curvature and Gaussian curvature can be evaluate by $$H = \frac{-1}{2}(EN - 2FM + GL)/W^2 \\ K = (LN - M^2)/W^2~~~~~~~~~~~~~~~~~~~~~$$ Therefore, the principal curvatures can be formulated in function of $p,p_x,p_{xx},p_y,p_{yy},y_{xy}$ by $$\kappa_1 = H - \sqrt{H^2 - K}\\ \kappa_2 = H + \sqrt{H^2 - K}$$ for any point in surface $M$.

But now I do not have idea to compute the principal directions corresponding to $\kappa_1$ and $\kappa_2$. Should I compute them by solving eigenvector of shape operator $S$? Thank you for your help.

• How could I do that? I am so confusing about the shape operator $S$. – sinoky Apr 1 '17 at 9:54
• Why the shape operator is not a 3 by 3 matrix in wiki page? I think that the shape operator is a mapping form $R^3$ to $R^3$. I am so confusing... – sinoky Apr 1 '17 at 10:14
• The shape operator should be thought of as mapping the tangent plane of your surface to the tangent plane of the sphere. Although both sit inside $\mathbb{R}^3$ they are both two dimensional. – phunfd Apr 1 '17 at 10:18
• I use the formula of shape operator $S$ in wiki and compute the solution of $(S - \kappa_i I)v = 0$, then I can get two 2 dimensional vectors $v = (v_1,v_2)$ corresponding $\kappa$. But our space is 3 dimensional, the principal direction must be 3D vector. Where is my mistake – sinoky Apr 1 '17 at 13:37