igllib 202_gaussian curvature

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the original tutor is like:

there are some points I wanna mention:

1. the discrete Gaussian curvature formula it listed is wrong. How? listen to me explain.

According to the paper "Discrete Differential-Geometry Operators for Triangulated 2-Manifolds" section 4.2, it says

In addition, the code also told us about that:

#include <igl/gaussian_curvature.h>#include <igl/massmatrix.h>#include <igl/invert_diag.h>#include <igl/readOFF.h>#include <igl/viewer/Viewer.h>#include <igl/jet.h>#include "tutorial_shared_path.h"int main(int argc, char *argv[]){  using namespace Eigen;  using namespace std;  MatrixXd V;  MatrixXi F;  igl::readOFF(TUTORIAL_SHARED_PATH "/bumpy.off",V,F);  VectorXd K;  // Compute integral of Gaussian curvature  igl::gaussian_curvature(V,F,K);  // Compute mass matrix  SparseMatrix<double> M,Minv;  igl::massmatrix(V,F,igl::MASSMATRIX_TYPE_DEFAULT,M);  igl::invert_diag(M,Minv);  // Divide by area to get integral average, note this line  K = (Minv*K).eval();  // Compute pseudocolor  MatrixXd C;  igl::jet(K,true,C);  // Plot the mesh with pseudocolors  igl::viewer::Viewer viewer;  viewer.data.set_mesh(V, F);  viewer.data.set_colors(C);  viewer.launch();}

2. I want to offer some material to help readers understand the geometrical meaning

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