旋转局部点时结果错误答案

【问题标题】：Wrong result when rotating a local point旋转局部点时结果错误
【发布时间】：2018-08-06 21:28:22
【问题描述】：

我已经设置了一个旋转矩阵，它似乎运行良好，但是当我围绕一个旋转点旋转一个局部点时，我无法获得正确的最终结果。我的代码：

glm::mat4 clsBone::Rotate( float a_Pitch, float a_Yaw, float a_Roll )
    {
        glm::mat4 l_M = m_MatrixHandler->GetRotationMatrix( );
        OutputDebugStringA( ( "Old Rotation Matrix: \n" +
                              std::to_string( l_M[ 0 ][ 0 ] ) + ", " + std::to_string( l_M[ 0 ][ 1 ] ) + ", " + std::to_string( l_M[ 0 ][ 2 ] ) + ", " + std::to_string( l_M[ 0 ][ 3 ] ) + "\n" +
                              std::to_string( l_M[ 1 ][ 0 ] ) + ", " + std::to_string( l_M[ 1 ][ 1 ] ) + ", " + std::to_string( l_M[ 1 ][ 2 ] ) + ", " + std::to_string( l_M[ 1 ][ 3 ] ) + "\n" +
                              std::to_string( l_M[ 2 ][ 0 ] ) + ", " + std::to_string( l_M[ 2 ][ 1 ] ) + ", " + std::to_string( l_M[ 2 ][ 2 ] ) + ", " + std::to_string( l_M[ 2 ][ 3 ] ) + "\n" +
                              std::to_string( l_M[ 3 ][ 0 ] ) + ", " + std::to_string( l_M[ 3 ][ 1 ] ) + ", " + std::to_string( l_M[ 3 ][ 2 ] ) + ", " + std::to_string( l_M[ 3 ][ 3 ] ) + "\n" ).c_str( ) );
        glm::mat4 l_RotMatrix = m_MatrixHandler->Rotate( a_Pitch, a_Yaw, a_Roll );
        l_M = l_RotMatrix;
        OutputDebugStringA( ( "New Rotation Matrix: \n" +
                              std::to_string( l_M[ 0 ][ 0 ] ) + ", " + std::to_string( l_M[ 0 ][ 1 ] ) + ", " + std::to_string( l_M[ 0 ][ 2 ] ) + ", " + std::to_string( l_M[ 0 ][ 3 ] ) + "\n" +
                              std::to_string( l_M[ 1 ][ 0 ] ) + ", " + std::to_string( l_M[ 1 ][ 1 ] ) + ", " + std::to_string( l_M[ 1 ][ 2 ] ) + ", " + std::to_string( l_M[ 1 ][ 3 ] ) + "\n" +
                              std::to_string( l_M[ 2 ][ 0 ] ) + ", " + std::to_string( l_M[ 2 ][ 1 ] ) + ", " + std::to_string( l_M[ 2 ][ 2 ] ) + ", " + std::to_string( l_M[ 2 ][ 3 ] ) + "\n" +
                              std::to_string( l_M[ 3 ][ 0 ] ) + ", " + std::to_string( l_M[ 3 ][ 1 ] ) + ", " + std::to_string( l_M[ 3 ][ 2 ] ) + ", " + std::to_string( l_M[ 3 ][ 3 ] ) + "\n" ).c_str( ) );
        glm::vec4 l_LocalPos = glm::vec4( m_EndJoint->m_Position - m_StartJoint->m_Position, 1 );
        OutputDebugStringA( ( "Old Local Pos: " + std::to_string( l_LocalPos.x ) + ", " + std::to_string( l_LocalPos.y ) + ", " + std::to_string( l_LocalPos.z ) + "\n" ).c_str( ) );
        glm::vec4 l_NewLocalPos = l_LocalPos * l_RotMatrix;
        OutputDebugStringA( ( "New Local Pos: " + std::to_string( l_NewLocalPos.x ) + ", " + std::to_string( l_NewLocalPos.y ) + ", " + std::to_string( l_NewLocalPos.z ) + "\n" ).c_str( ) );

        return l_RotMatrix;
    }

glm::mat4 clsMatrixHandler::Rotate( float a_Pitch, float a_Yaw, float a_Roll )
{
    glm::mat4 l_Rotx;
    glm::mat4 l_Roty;
    glm::mat4 l_Rotz;
    PitchYawRollToXYZMatrices( a_Pitch, a_Yaw, a_Roll, l_Rotx, l_Roty, l_Rotz );
    m_PitchYawRolls.push_back( glm::vec3( a_Pitch, a_Yaw, a_Roll ) );
    glm::mat4 l_RotationMatrix = l_Rotx * l_Roty * l_Rotz;
    m_RotationMatrix *= l_RotationMatrix;
    m_TransformMatrix = m_RotationMatrix * m_TranslationMatrix;
    return m_RotationMatrix;
}

结果：

Old Rotation Matrix: 
1.000000, 0.000000, 0.000000, 0.000000
0.000000, -0.000000, 1.000000, 0.000000
0.000000, -1.000000, -0.000000, 0.000000
0.000000, 0.000000, 0.000000, 1.000000
New Rotation Matrix: 
0.707107, 0.707107, 0.000000, 0.000000
0.000000, -0.000000, 1.000000, 0.000000
0.707107, -0.707107, -0.000000, 0.000000
0.000000, 0.000000, 0.000000, 1.000000
Old Local Pos: 0.000000, -40.000000, -0.000002
New Local Pos: -28.284271, 0.000000, 28.284271

新的局部位置怎么可能向对角线方向旋转 90 度，而旋转矩阵在其局部 y 轴（世界 z 轴）上旋转 45 度？

【问题讨论】：

标签： rotational-matrices

【解决方案1】：

看来我在这里犯了一个错误。我将本地位置与旋转矩阵相乘，但我应该以相反的方式相乘：

    glm::vec4 l_NewLocalPos = l_RotMatrix * l_LocalPos;

这将输出：

Old Rotation Matrix: 
1.000000, 0.000000, 0.000000, 0.000000
0.000000, -0.000000, 1.000000, 0.000000
0.000000, -1.000000, -0.000000, 0.000000
0.000000, 0.000000, 0.000000, 1.000000
New Rotation Matrix: 
0.707107, 0.707107, 0.000000, 0.000000
0.000000, -0.000000, 1.000000, 0.000000
0.707107, -0.707107, -0.000000, 0.000000
0.000000, 0.000000, 0.000000, 1.000000
Old Local Pos: 0.000000, -40.000000, -0.000002
New Local Pos: 0.000000, 0.000000, -40.000000

这是有道理的：从恒等旋转矩阵中，我首先向下俯仰，将局部点从 (0, -40, 0) 移动到 (0, 0, -40)，然后我偏航 45 度，但是由于局部点在yaw 旋转轴（局部 y 轴，世界 z 轴），该点不会移动。

【讨论】：