Quaternion as 3D Rotation Hand

Quaternion algebra was introduced by Hamilton in 1843. It is powerful three dimensional complex number system. This article will present how to use quaternion  as rotation tools which is very useful for the calculation involving three-dimensional rotation in 3D computer graphics.

In order to match with OpenGL, this article apply the right hand rule to explain quaternion . If you come across another article which based on the left hand rule , do not be surprised to see that the terms are different to those described in this article.

Quaternion multiplication of basis element

  Q  =  w  +  x i   +   yj   +  zk  

   where  i , j , k  is complex number   w,x,y,z  are real number

     i²  =    j²  =   k²  =   i j k   =   -1

    ij = k  ;   ji = -k

    jk = i  ;   kj = -i

    ki =j   ;  ik = -j

   The  x,y,z  is rotation axis  and   w  is some thing which relate to the rotation angle required.

Quaternion Product

      define qutaternion   Q₁  and Q₂

      Q₁  =  w₁  +  x₁ i   +   y₁ j   +  z₁ k     =   w₁ +  v₁     where  v₁  = (  x₁ i   +   y₁ j   +  z₁ k  )

      Q₂  =  w₂ +  x₂ i    +   y₂ j   +  z₂ k     =   w₂ +  v₂     weher  v₂  =  ( x₂ i   +   y₂ j   +  z₂ k )

      Q₁*Q₂  =  w₁w₂ +  w₁v₂ +w₂v₁ + v₁v₂

      v₁v₂    =    - ( x₁x₂ + y₁y₂ +z₁z₂ ),     ==>   -  ( v₁.v₂ )  dot product

                   
                        + (y₁z₂  - z₁y₂)  i ,

                        + (z₁x₂  - x₁z₂)  j ,             ==>       ( v₁ x v₂ )      cross product

                        + (x₁y₂  - y₁x₂)  k

       Q₁*Q₂    =   w₁w₂ + w₁v₂ +w₂v₁ - v₁.v₂ + v₁ x v₂

                      = ( w₁w₂  - v₁.v₂)    +  (w₁v₂ +w₂v₁  + v₁ x v₂)

                      = ........

 
                   replace the terms with

                   i2 =   j2 =  k2 =   i j k   =   -1
 
                   ij = k  ;   ji = -k
 
                   jk = i  ;   kj = -i
 
                   ki =j   ;  ik = -j