均值重心坐标Mean Value Coordinates In 3D代码实现

c++实现：

关于2维中均值重心坐标的实现可参考我的另一篇博客：均值重心坐标Mean Value Coordinates In 2D代码实现

/*
/// @brief      mean value coordinates in 3D
/// @details    
/// @param[in]  p: 
/// @return     
/// @attention  
*/ 
VectorXd MyMesh::meanValueCoordinates(Vector3d p, bool &postiveity)
{
    postiveity = true;
    //this->faces = this->fixMesh().faces;   // repair the faces index, so that the all normal vector of faces outward (inward)
    VectorXd mvc_coor = VectorXd::Zero(this->numberOfVertices);

    // get projection vertices on unit sphere
    Matrix<double,3,Dynamic> pro_vertices;

    pro_vertices.resize(3, this->numberOfVertices);
    for (int i = 0; i < this->numberOfVertices; ++i)
    {
        double r = (this->vertices.col(i) - p).norm();
        pro_vertices(0, i) = (this->vertices(0, i) - p(0)) / r;
        pro_vertices(1, i) = (this->vertices(1, i) - p(1)) / r;
        pro_vertices(2, i) = (this->vertices(2, i) - p(2)) / r;
    }

    Matrix<double, 3, 3> T;
    Vector3d n0, n1, n2;
    double beta0, beta1, beta2;
    for (int i = 0; i < this->numberOfFaces; ++i)
    {
        int i0, i1, i2;     // index of vertices on the current face
        i0 = this->faces(0, i);
        i1 = this->faces(1, i);
        i2 = this->faces(2, i);

        T.col(0) = pro_vertices.col(i0);
        T.col(1) = pro_vertices.col(i1);
        T.col(2) = pro_vertices.col(i2);

        n0 = pro_vertices.col(i0).cross(pro_vertices.col(i1)).normalized();
        n1 = pro_vertices.col(i1).cross(pro_vertices.col(i2)).normalized();
        n2 = pro_vertices.col(i2).cross(pro_vertices.col(i0)).normalized();

        // angle
        beta0 = acos(pro_vertices.col(i0).dot(pro_vertices.col(i1)));
        beta1 = acos(pro_vertices.col(i1).dot(pro_vertices.col(i2)));
        beta2 = acos(pro_vertices.col(i2).dot(pro_vertices.col(i0)));

        mvc_coor[i0] += (beta1 + beta0*n0.dot(n1) + beta2*n2.dot(n1)) / (2 * pro_vertices.col(i0).dot(n1));
        mvc_coor[i1] += (beta2 + beta1*n1.dot(n2) + beta0*n0.dot(n2)) / (2 * pro_vertices.col(i1).dot(n2));
        mvc_coor[i2] += (beta0 + beta2*n2.dot(n0) + beta1*n1.dot(n0)) / (2 * pro_vertices.col(i2).dot(n0));
    }
    for (int i = 0; i < this->numberOfVertices; ++i)
    {
        mvc_coor[i] /= (this->vertices.col(i) - p).norm();
    }
    mvc_coor /= mvc_coor.sum();
    postiveity = mvc_coor.minCoeff() >= 0;
    return mvc_coor;
}

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