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

**** Finally you will get ****

   Q₁*Q₂   =        ( w₁w₂  - x₁x₂ - y₁y₂ -z₁z₂ )                  

                        + ( w₁x₂ + w₂x₁ +y₁z₂ -z₁y₂ )  i

                        + ( w₁y₂ + w₂y₁ +z₁x₂ -x₁z₂ )  j

                        + ( w₁z₂ + w₂z₁ +x₁y₂ -y₁x₂ )  k

          @!!       Q₁*Q₂   !=   Q₂*Q₁

Normalise  Quaternion

      Q  =  w  +  x i   +   y j   +  z k    

      Q_mag  =  ( w² + x² + y² + z²  )^1/2

      Q_normal   =  w/Q_mag +  (x/Q_mag)  i   +   (y/Q_mag)  j   + ( z/Q_mag)   k    

Conjugate  Quaternion

     Q  =  w +  x  i   +   y  j   +  z  k   

     Q_conjugate  =  w  -  x  i  -   y j   -  z  k

HOW TO USE QUATERNION AS ROTATION HAND

     

   
     Suppose that you want to rotate the object  located at point     P =  (x₀,y₀,z₀) 
     with angle  θ  around   Axis   =  a_x x + a_y y + a_z z

   Step 1 ::  Build  quaternion   Q
   
      Q  =  w +  x  i   +   y  j   +  z  k    
   
     where  ::

      w  =  cos(θ/2)

      x   =  a_x   * sin(θ/2)

      y   =  a_y   * sin(θ/2)

      z   =  a_z   * sin(θ/2)

   
      Then your quaternion rotation hand is 

      Q  =  cos(θ/2)  +  (a_x *sin(θ/2)) i  + (a_y *sin(θ/2)) j   +  (a_z*sin(θ/2))  k  

   Step 2 ::  Normalise  quaternion   Q  => Q_normal

   Step 3 ::  Conjugate quaternion   Q_normal   = > Qconjugate

   Step 4 :: Transform object coordinate at point  P(x₀,y₀,z₀)  into quaternion format  Q_p
       w  =  0
       x   =  x₀
       y   =  y₀
       z   =  z₀
      Q_p  =  0 + x₀ i + y₀ j+ z₀ k

    Step 5 :: Get final coordinate from equation

    Q_final  = Q_normal  * Qp  *  Qconjugate  =     w +  x_f  i   +   y_f  j   +  z_f  k  

    where  x_f,y_f,z_f    is  new coordinate of object resulting from rotation.