anchor.diff:  
   This representation means if we fully understand flip, which is easier to visualize, we can fully understand rotation and quaternions, since any quaternion can be broken down to flips.
   
  -== Quaternion and Flip
  +=== Quaternion and Flip
   
   Now let’s recall the quaternion vector form stem:[q=(sin⁡\frac{θ}{2}\vec{v},cos⁡\frac{θ}{2})]. With the discussion of flips above, you can almost immediately see why we are using stem:[\frac{θ}{2}] here.
   
  
positive.code: :stem: latexmath
  :figure-caption!:
  
  == Before We Start
  
  Quaternion is widely used in game engines to represent 3D rotation. As a game engineer you might be using quaternion explicitly or implicitly in your daily work, but do you really understand what is going on under the hood when you are calling “rotate a vector” or “combine two rotations”? Why rotating a vector stem:[v] by quaternion stem:[q] is calculated by a “sandwich” multiplication: stem:[qvq^{-1}] ? Why rotating by quaternion stem:[q_1] then stem:[q_2] is in the reversed order: stem:[{q_2}{q_1}], and can you visualize the result rotation axis and angle?
  
  
negative.code: 
  From now on, any quaternion stem:[q] used in this article is by default a unit quaternion, and we will use stem:[q] to describe rotations.
  
  === Rotation and Flip
  
  Now let’s forget quaternion for a minute, and focus on the nature of rotations. This part is the key to understand quaternion calculation in an easier way.
  
  