DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

82 Chapter 3. Surfaces: Further Topicsinterior vertexinterior edge∆ µ∆ λFigure 1.8boundary edgesboundary vertexinterior/boundary edgeson the boundary, edges in the interior, and edges that join a boundary vertex to an interior vertex;we denote the respective numbers of these by E b , E i , and E ib .Now observe that∫∫m∑∫∫KdA = KdAR∆ λsince all the orientations are compatible, and∫κ g ds =∂Rλ=1m∑∫λ=1∂∆ λκ g dsbecause the line integrals over interior and interior/boundary edges cancel in pairs. Let ɛ λj , j =1, 2, 3, denote the exterior angles of the “triangle” ∆ λ . Then, applying Theorem 1.5 to ∆ λ ,wehave∫∫∫3∑κ g ds + KdA+ ɛ λj =2π,∂∆ λ ∆ λand now, summing over the triangles, we obtain∫ ∫∫κ g ds + KdA+∂MMm∑λ=1 j=1j=13∑ɛ λj =2πm =2πF .Now we must do some careful accounting: Letting ι λj denote the respective interior angles oftriangle ∆ λ ,wehave(∗)∑ɛ λj =∑(π − ι λj )=π(2E i + E ib ) − 2πV iinteriorverticesinteriorverticesinasmuch as each interior edge contributes two interior vertices, whereas each interior/boundaryedge contributes just one, and the interior angles at each interior vertex sum to 2π. Next,(∗∗)∑boundaryverticesɛ λj = πE ib +l∑ɛ k .k=1