void part2()
{
    std::cout << "Part 2: Simple ray tracer, one sphere with orthographic projection" << std::endl;

    const std::string filename("part2.png");
    MatrixXd C = MatrixXd::Zero(800,800); // Store the color
    MatrixXd A = MatrixXd::Zero(800,800); // Store the alpha mask

    // The camera is orthographic, pointing in the direction -z and covering the unit square (-1,1) in x and y

    Vector3d origin(-1,1,1);
    Vector3d x_displacement(2.0/C.cols(),0,0);
    Vector3d y_displacement(0,-2.0/C.rows(),0);

    // Single light source
    const Vector3d light_position(-1,1,1);

    for (unsigned i=0;i<C.cols();i++)
    {
        for (unsigned j=0;j<C.rows();j++)
        {
            // Prepare the ray
            Vector3d ray_origin = origin + double(i)*x_displacement + double(j)*y_displacement;
            Vector3d ray_direction = RowVector3d(0,0,-1);

            // Intersect with the sphere
            // NOTE: this is a special case of a sphere centered in the origin and for orthographic rays aligned with the z axis
            Vector2d ray_on_xy(ray_origin(0),ray_origin(1));
            const double sphere_radius = 0.9;

            if (ray_on_xy.norm()<sphere_radius)
            {
                // The ray hit the sphere, compute the exact intersection point
                Vector3d ray_intersection(ray_on_xy(0),ray_on_xy(1),sqrt(sphere_radius*sphere_radius - ray_on_xy.squaredNorm()));

                // Compute normal at the intersection point
                Vector3d ray_normal = ray_intersection.normalized();

                // Simple diffuse model
                C(i,j) = (light_position-ray_intersection).normalized().transpose() * ray_normal;

                // Clamp to zero
                C(i,j) = max(C(i,j),0.);

                // Disable the alpha mask for this pixel
                A(i,j) = 1;
            }
        }
    }


    // Save to png
    write_matrix_to_png(C,C,C,A,filename);

}