Differential Geometry and Shape Analysis - Mathematical Tools for Shape Analysis and Description - page 30

Figure 4.3: Normal vector and principal curvatures on a saddle point.
On the basis of the value of the Gaussian curvature (either positive or negative or zero) it is
possible to classify a point as elliptic, hyperbolic or parabolic; see examples in Figure 4.4 .
Figure 4.4: An elliptic point corresponds to K > 0 (a), an hyperbolic point (b) has K < 0 , while for
parabolic points (c) the Gaussian curvature is zero; if also HD0 the surface is locally a plane.
e mean curvature is a purely extrinsic property, that is, it depends on how the surface
is embedded in the Euclidean space R 3 . On the contrary, whereas the definition of Gaussian
curvature is extrinsic in that it uses the surface's embedding in R 3 , that is normal vectors and
normal planes, Gaussian curvature is in fact an intrinsic property of the surface, which does not
depend on its particular embedding, but only on the Riemannian (intrinsic) metric of the surface.
is surprising result is the celebrated eorema Egregium (Latin expression meaning “remark-
able theorem”) which Gauss found while concerned with geographic surveys and map making:
the Gaussian curvature of a smooth surface embedded in R 3 is invariant under isometric defor-
mations of the surface (cf. Sect. 6.1 ). In other words, Gauss's eorema Egregium states that
Gaussian curvature of a surface can be determined from the measurements of length on the sur-
face itself: intuitively, this means that an ant living on the surface could determine the Gaussian
curvature while walking on the surface, without any reference to the external space.
e Gaussian curvature, as an intrinsic invariant, can be given entirely in terms of the first
fundamental form. A simpler formula is given in terms of both the first and second fundamental