$\mu$ : Sfm Points

$\Sigma$ : Covariance matrix

• $q$ : Quarternion(for Rotation)

  $q = w+xi+yj+zk \\s.t.~~ w = cos(\theta/2)\\ ~~~~~~~~x = u_xsin(\theta/2)\\ ~~~~~~~~y = u_ysin(\theta/2)\\ ~~~~~~~~z = u_zsin(\theta/2) \\

  
  $v'=qvq^* ~~~s.t.~~ q^*~: ~conjugate ~quaternion\\\\
  let~v=[1~~0~~0] \\because ~it~is~\\mathbf{x-axis}\\\\
  since~q~is~on~complex~space,~v=0+1*i+0*j+0*k=i\\\\
  v'=(w+xi+yj+zk)v(w-xi-yj-zk)\\\\
  ~~~~=(wi+xii+yji+zki)(w-xi-yj-zk)\\\\
  ~~~~=(wi-x-yk+zj)(w-xi-yj-zk)\\\\
  ~~~~=...\\\\
  ~~~~=(w^2+x^2−y^2−z^2)i+(2xy+2wz)j+(2xz−2wy)k\\\\
  ~~~~=(1−2(y^2+z^2))i+(2xy+2wz)j+(2xz−2wy)k\\\\
  ~~~~\\because  ~w^2+x^2+y^2+z^2=1~(q~is~unit~vector).$
  
  

• $s$ : Scale(for Magnitude)

  $S = \begin{bmatrix} exp(s_x)&0&0\\0&exp(s_y)&0\\0&0&exp(s_z)\end{bmatrix}\\s.t.~s_{sth} : scalability~of~each~axis$

$C$ : Color based on SH coefficient

$\frac{\partial Loss}{\partial C} = \frac{\partial Loss}{\partial C}\frac{\partial C}{\partial k}$

$\alpha$ : Opacity(for Transmittance)

$\alpha'=\alpha G(x)$