Eigen入门（2）

主要根据视觉SLAM十四讲配套代码编写
包含头文件
#include
#include

定义单位矩阵：

Matrix3d rotation_matrix = Matrix3d::Identity();
cout<<rotation_matrix<<endl;

1 0 0
0 1 0
0 0 1

定义旋转向量

AngleAxisd rotation_vector(M_PI / 4, Vector3d(0, 0, 1));  //沿Z 轴旋转45度
cout.precision(3);
cout << "rotation matrix =\n" << rotation_vector.matrix() << endl;   //用matrix()转换成矩阵

rotation matrix =
0.707 -0.707 0
0.707 0.707 0
0 0 1

由旋转向量定义旋转矩阵

 rotation_matrix = rotation_vector.toRotationMatrix();

定义一个坐标

Vector3d v(1, 0, 0);

对坐标进行旋转
1、利用旋转向量

Vector3d v_rotated = rotation_vector * v;
cout << "(1,0,0) after rotation (by angle axis) = " << v_rotated.transpose() << endl;

2、利用旋转矩阵

v_rotated = rotation_matrix * v;
cout << "(1,0,0) after rotation (by matrix) = " << v_rotated.transpose() << endl;

(1,0,0) after rotation (by angle axis) = 0.707 0.707 0
(1,0,0) after rotation (by matrix) = 0.707 0.707 0
将旋转向量转换成欧拉角

Vector3d euler_angles = rotation_matrix.eulerAngles(2, 1, 0); // ZYX顺序，即yaw-pitch-roll顺序
  cout << "yaw pitch roll = " << euler_angles.transpose() << endl;

定义欧式变换矩阵（4*4，即包括旋转和平移，）

Isometry3d T = Isometry3d::Identity();                // 虽然称为3d，实质上是4＊4的矩阵
  T.rotate(rotation_vector);                                     // 按照rotation_vector进行旋转
  T.pretranslate(Vector3d(1, 3, 4));                     // 把平移向量设成(1,3,4)
  cout << "Transform matrix = \n" << T.matrix() << endl;

Transform matrix =
0.707 -0.707 0 1
0.707 0.707 0 3
0 0 1 4
0 0 0 1
用变换矩阵进行坐标变换

Vector3d v_transformed = T * v;                              // 相当于R*v+t
  cout << "v tranformed = " << v_transformed.transpose() << endl;

v tranformed = 1.71 3.71 4

四元数

1、旋转向量定义四元数

Quaterniond q = Quaterniond(rotation_vector);
  cout << "quaternion from rotation vector = " << q.coeffs().transpose()
       << endl;   // 请注意coeffs的顺序是(x,y,z,w),w为实部，前三者为虚部

quaternion from rotation vector = 0 0 0.383 0.924
2、旋转矩阵定义四元数

 q = Quaterniond(rotation_matrix);
  cout << "quaternion from rotation matrix = " << q.coeffs().transpose() << endl;

quaternion from rotation matrix = 0 0 0.383 0.924

利用四元数进行旋转

 v_rotated = q * v; // 注意数学上是qvq^{-1}
  cout << "(1,0,0) after rotation = " << v_rotated.transpose() << endl;
  // 用常规向量乘法表示，则应该如下计算
  cout << "should be equal to " << (q * Quaterniond(0, 1, 0, 0) * q.inverse()).coeffs().transpose() << endl;

(1,0,0) after rotation = 0.707 0.707 0
should be equal to 0.707 0.707 0 0
上述完整代码：

#include 
#include 

using namespace std;

#include 
#include 

using namespace Eigen;

// 本程序演示了 Eigen 几何模块的使用方法

int main(int argc, char **argv) {

  // Eigen/Geometry 模块提供了各种旋转和平移的表示
  // 3D 旋转矩阵直接使用 Matrix3d 或 Matrix3f  Identity 单位矩阵
  Matrix3d rotation_matrix = Matrix3d::Identity();
  cout<<rotation_matrix<<endl;
  // 旋转向量使用 AngleAxis, 它底层不直接是Matrix，但运算可以当作矩阵（因为重载了运算符）
  AngleAxisd rotation_vector(M_PI / 4, Vector3d(0, 0, 1));     //沿 Z 轴旋转 45 度
  cout.precision(3);
  cout << "rotation matrix =\n" << rotation_vector.matrix() << endl;   //用matrix()转换成矩阵
  // 也可以直接赋值
  rotation_matrix = rotation_vector.toRotationMatrix();
  // 用 AngleAxis 可以进行坐标变换
  Vector3d v(1, 0, 0);
  Vector3d v_rotated = rotation_vector * v;
  cout << "(1,0,0) after rotation (by angle axis) = " << v_rotated.transpose() << endl;
  // 或者用旋转矩阵
  v_rotated = rotation_matrix * v;
  cout << "(1,0,0) after rotation (by matrix) = " << v_rotated.transpose() << endl;

  // 欧拉角: 可以将旋转矩阵直接转换成欧拉角
  Vector3d euler_angles = rotation_matrix.eulerAngles(2, 1, 0); // ZYX顺序，即yaw-pitch-roll顺序
  cout << "yaw pitch roll = " << euler_angles.transpose() << endl;

  // 欧氏变换矩阵使用 Eigen::Isometry
  Isometry3d T = Isometry3d::Identity();                // 虽然称为3d，实质上是4＊4的矩阵
  T.rotate(rotation_vector);                                     // 按照rotation_vector进行旋转
  T.pretranslate(Vector3d(1, 3, 4));                     // 把平移向量设成(1,3,4)
  cout << "Transform matrix = \n" << T.matrix() << endl;

  // 用变换矩阵进行坐标变换
  Vector3d v_transformed = T * v;                              // 相当于R*v+t
  cout << "v tranformed = " << v_transformed.transpose() << endl;

  // 对于仿射和射影变换，使用 Eigen::Affine3d 和 Eigen::Projective3d 即可，略

  // 四元数
  // 可以直接把AngleAxis赋值给四元数，反之亦然
  Quaterniond q = Quaterniond(rotation_vector);
  cout << "quaternion from rotation vector = " << q.coeffs().transpose()
       << endl;   // 请注意coeffs的顺序是(x,y,z,w),w为实部，前三者为虚部
  // 也可以把旋转矩阵赋给它
  q = Quaterniond(rotation_matrix);
  cout << "quaternion from rotation matrix = " << q.coeffs().transpose() << endl;
  // 使用四元数旋转一个向量，使用重载的乘法即可
  v_rotated = q * v; // 注意数学上是qvq^{-1}
  cout << "(1,0,0) after rotation = " << v_rotated.transpose() << endl;
  // 用常规向量乘法表示，则应该如下计算
  cout << "should be equal to " << (q * Quaterniond(0, 1, 0, 0) * q.inverse()).coeffs().transpose() << endl;

  return 0;
}