The total curvature C of a genus -0 polyhedron like a cube is obviously 4 . It is not very usual to find adequate generalizations of the GB theorem for surfaces with impulse curvature at points not belonging to the boundary-for instance, polyhedra. This result may be understood very clearly from the area ⌫ en- closed by the Gauss map of Fig. For a compact regular surface homomorphic to a sphere, ϭ 2, C ϭ ͐ Kd ϭ 4 , and the total, or integral, curvature is invariant. By triangulating a general surface one gets the global version of the Gauss–Bonnet formula, which states that if r is a piecewise regular curve bounding a region S of an oriented surface, then ͐ S Kd ϩ ͐ r g ds ϩ ͚ i ␣ i ϭ 2 , where is the Euler–Poincar ́ characteristic of the surface, an invariant that may be obtained by the Euler formula F Ϫ E ϩ V ϭ , where F, E, and V are the number of faces, edges, and vertices of the triangulation. If the boundary is only piecewise regular, there are impulse curvatures ␣ i at the i th junction, and the theorem generalizes to ͐ S Kd ϩ ͐ r g ds ϩ ͚ i ␣ i ϭ 2 . The first term may be identified with the area ⌫ of Fig. In its local version, it states that for a simply connected oriented surface patch S with a regular curve r ( s ) as its boundary, ץ S ϭ r, ͐ S Kd ϩ ͐ r g ds ϭ 2 . Of utmost importance is the Gauss–Bonnett ͑ GB ͒ theorem, one of the deepest results in the differential geometry of surfaces. The two invariants of the second form are the mean curvature H ϭ 1/2( 1 ϩ 2 ) and the Gauss curvature K ϭ 1 2. The minimum, , and maximum, , normal curvatures are the principal curvatures of the surface at a point and their orthogonal tangent directions its principal directions. This is the osculating paraboloid that we will use in the following. The elevation of a surface above its tangent plane at x ( u ) is given, up to second-order terms, by h ( v ) ϭ 1/2 ͚ i j i j ( v Ϫ u ) i ( v Ϫ u ) j. These relations are also visualized in Fig. The projection of r Љ ( s ) onto the tangent plane is called the geodesic curvature g. N, the normal curvature of the curve, N.If t is the unit tangent to an arclength parametrized curve r ( s ) on the surface, ( t, t ) equals r Љ ( s ) Its components in the u coordinates are i j ϭ x i j Minus the tangential directional derivative of the unit normal is a symmetric bilinear form, the Weingarten tensor or second fundamental form, which measures the change of the tangent plane when moving onto the surface. The map N : U → S 2 of the normal field onto the unit sphere is called the Gauss map of the surface and is sketched in Fig. If it is possible to construct a differentiable unit normal field over the whole surface, we say the surface is ori- entable. The tangent plane is the affine space x ( p ) ϩ T p x, and the unit vector field N ϭ ( x 1 ϫ x 2 )/ ͉ x 1 ϫ x 2 ͉ is called the unit normal. The tangent space of x at p ʚ U, T p x, is the 2D subspace generated by the vectors x 1 ( p ) and x 2 ( p ). A parametric surface patch is a smooth function x ( u 1, u 2 ): U → R 3, where U ʚ R 2 is an open set and the Jaco- bian of x is nonsingular. The reader may check that if a curve is only piecewise regular, the previous expression is still valid if we assume that an additive impulse curvature, or angle defect, is present at each angular point, r ang, and equal to the exterior angle change between the left and right tangents of the curve at r ang. As a global result, it is easy to show that ␣ ϭ ͐ b a ds is the exterior angle change between the tangents of the curve at points r ( a ) and r ( b ). The scalar defined by b Ј ϭϪ n Ј is called the torsion, and the fundamental theorem of the local theory of curves states that given the functions ( s ) Ͼ 0 and ( s ), there is only one curve, defined up to a given rigid motion, whose curvature and torsion are given by those functions. The vector b ϭ t ϫ n is the binormal, the t - n plane the osculator plane, and the t, n, b basis the Frenet reference frame of the curve at a given point ͑ see Fig. The module of the vector r Љ is the curvature scalar and the unit vector n ϭ Ϫ 1 t Ј, the principal normal. are critical or singular points, and a curve without singular points is named regular.
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