anchor.diff:  
   === Rotation along X/Y/Z Axis
   
  -We can now go back to the problem I mentioned at the very beginning: we need an object to face its opposite direction. More clearly we have an object with rotation stem:[q=((x,y,z),w)], and we want to flip it along local Z axis, that is rotate it stem:[180^{\circ}] along its local Z axis. This extra rotation is denoted as stem:[q'=((0,0,\sin\frac{180^{\circ}}{2},\cos\frac{180^{\circ}}{2})=((0,0,1),0)]. Based on local rotation composition we proved in previous section, the result is 
  +We can now go back to the problem I mentioned at the very beginning: we need an object to face its opposite direction. More clearly we have an object with rotation stem:[q=((x,y,z),w)], and we want to flip it along local Z axis, that is rotate it stem:[180^{\circ}] along its local Z axis. This extra rotation is denoted as stem:[q'=((0,0,\sin\frac{180^{\circ}}{2}),\cos\frac{180^{\circ}}{2})=((0,0,1),0)]. Based on local rotation composition we proved in previous section, the result is 
   
   [stem]
   ++++
   q_Z=qq'=((x,y,z),w)((0,0,1),0)=((y,-x,w),-z)
   ++++
   
  -If we generalize the angle to stem:[θ], then stem:[q'=((0,0,\sin\frac{θ}{2},\cos\frac{θ}{2})], then the result is:
  +If we generalize the angle to stem:[θ], then stem:[q'=((0,0,\sin\frac{θ}{2}),\cos\frac{θ}{2})], then the result is:
   
   [stem]
   ++++
   \begin{align*}
   q_{(Z,θ)}=qq'&=((x,y,z),w)((0,0,\sin\frac{θ}{2},\cos\frac{θ}{2})\\
   &=((x,y,z),w)(((0,0,0),1)\cos\frac{θ}{2}+((0,0,1),0)\sin\frac{θ}{2})\\
  -&=((x,y,z),w)\cos\frac{θ}{2}+((y,-x,w),-z)\sin\frac{θ}{2})\\
  +&=((x,y,z),w)\cos\frac{θ}{2}+((y,-x,w),-z)\sin\frac{θ}{2}\\
   \end{align*}
   ++++  
   
  
positive.code: [stem]
  ++++
  \begin{align*}
  q_{(X,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(w,z,-y,-x)\sin\frac{θ}{2})\\
  q_{(Y,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(-z,w,x,-y)\sin\frac{θ}{2})\\
  q_{(Z,θ)}=(x,y,z,w)\cos\frac{θ}{2}+(y,-x,w,-z)\sin\frac{θ}{2})\\
  \end{align*}
  ++++  
  
  
negative.code: q_Z=q'q=((0,0,1),0)((x,y,z),w)=((-y,x,w),-z)
  ++++
  
  We can use the same method to generalize the angle to stem:[θ], and let stem:[q'=((0,0,\sin\frac{θ}{2}),\cos\frac{θ}{2})],
  
  [stem]
  ++++
  