By Jacob T Schwartz

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6. If K = 0 and not both principal curvatures are zero then we have a parabolic point, and finally if both principal curvatures are zero then we have a planar point. On a negatively curved surface there are two directions where the normal curvature is zero and they are called the asymptotic directions. The mean curvature is the mean value of the principal curvatures, κ1 + κ2 − tr W tr(I−1I ) = = 2 2 2 1 g11 b22 − 2g12 b12 + g22 b11 . 7 The Gaußian and Mean Curvature 53 Fig. 6 On the left a positively curved surface, on the right a negatively curved surface Fig.

If this gives f1 , . . , fk then the rest of the basis vectors are chosen as an orthonormal basis for span{f1 , . . , fk }⊥ . In terms of matrices it has the following formulation. 12 Let A be an m × n matrix and let k = min{m, n}. Then A can be decomposed as A = UΣVT , where U is an orthogonal m × m matrix, V is an orthogonal n × n matrix, and Σ is a diagonal matrix with non zero elements σ1 ≥ σ1 ≥ · · · ≥ σk ≥ 0 in the diagonal. The singular values are the square root of the eigenvalues of AT A, which is a positive semi definite symmetric matrix.

6 Let h be the signed distance function for a surface S and let e1 , e2 , e3 be pairwise orthogonal eigenvectors for the Hessian H (h) such that e3 = ∇h. The principal directions for S are then e1 and e2 and if λ1 , λ2 , λ3 are the eigenvalues for H (h), then λ3 = 0 and the principal curvatures are κ1 = −λ1 and κ2 = −λ2 . We find in particular that the mean curvature is M= λ1 + λ2 tr H (h) 1 ∂ 2h ∂ 2h ∂ 2h κ1 + κ2 =− =− =− + + 2 , 2 2 2 2 ∂x 2 ∂y 2 ∂z 60 3 Differential Geometry and the Gaußian curvature is K = κ1 κ2 = λ1 λ2 = ∂2h ∂x 2 ∂2h ∂y∂x ∂2h ∂x∂y ∂2h ∂y 2 + ∂2h ∂x 2 ∂ 2h ∂z∂x ∂2h ∂x∂z ∂2h ∂z2 + ∂2h ∂y 2 ∂2h ∂y∂z ∂2h ∂y∂z ∂2h ∂z2 .