用四元數圍繞一個軸旋轉矢量

Question

我想學習三維程序設計，現在我試圖瞭解如何使用四元數來圍繞一個軸旋轉矢量。

據我所知，要將一個向量v圍繞軸a旋轉，將兩個向量轉換爲四元后，我們將v乘以a，然後乘以a的共軛。 （0,0,1）（或0,0，0,0,0）（0,0,0）（0,0,0） -1，取決於旋轉的方向）。

我沒有得到我期待的輸出。 下面是代碼：

int main() 
    { 
     //I want to rotate this vector about the x axis by PI/2 radians: 
     Quaternion v(0, 1, 0, 0); 
     v.normalize(); 


     float angle = PI/2.0f; 
     float cos = math::cos(angle/2.0f); 
     float sin = math::sin(angle/2.0f); 

     Quaternion q(1.0f*sin, 0.0f*sin, 0.0f*sin, cos); 

     std::cout << "q not normalized = " <<"\t"<< q.x << " " << q.y << " "  << q.z << " " << q.w << std::endl; 

     q.normalize(); 

     std::cout << "q normalized = " <<"\t\t"<< q.x << " " << q.y << " " << q.z << " " << q.w << std::endl; 
     std::cout << std::endl; 

     Quaternion r; 


     //I multiply the vector v by the quaternion v, then I multiply by the conjugate. 
     r = q * v; 
     //do I need to normalize here? 
     r = r * q.conjugate(); 
     //and here? 


     //shouldn't the resulting vector be 0,0,1? 

     std::cout << "r not normalized = " << "\t" << r.x << " " << r.y << " " << r.z << " " << r.w << std::endl; 
     r.normalize(); 

     std::cout << "r normalized = " << "\t\t" << r.x << " " << r.y << " " << r.z << " " << r.w << std::endl; 
     std::cout << std::endl; 

     system("pause"); 
     return 0; 
    } 

，這裏是輸出：

。q沒有歸一化，這是相同的爲q歸一化： X = 0.707107，Y = 0，Z = 0，W = 0.707107

R值不要歸一化： X = 0.707107，Y = 0，Z = 1，W = -2.12132 ř標準化： X = 0.288675，Y = 0，Z = 0.408248，W = -0.866025

我究竟做錯了什麼？ 我甚至能理解這個過程中的任何事情嗎？

Answer 1

基本上與角度90度旋轉沿x軸（1,0,0）的向量，使用下面的方法，這既適用於歐拉和四元數

| 1 0   0 | | 0 |  | 0 | 
| 0 cos90 -sin90 | * | 1 | = | 0 | 
| 0 sin90 cos90 | | 0 |  | 1 | 

閱讀關於旋轉矩陣http://en.wikipedia.org/wiki/Rotation_matrix