Homogeneous 4x4 matrices

Add a 4th coordinate:

P = [ x  
      y  
      z  
      1 ]

Now you can include translation in the matrix.

Translation matrix

T(dx, dy, dz) =  
[ 1  0  0  dx  
  0  1  0  dy  
  0  0  1  dz  
  0  0  0   1 ]

Rotation in 4×4 form (example: Z)

Rz(θ) =  
[ cos(θ)  -sin(θ)  0  0  
  sin(θ)   cos(θ)  0  0  
    0         0    1  0  
    0         0    0  1 ]

Rotate around arbitrary point (clean version)

Instead of manual translate/rotate/translate, combine matrices:

M = T(cx, cy, cz) * R * T(-cx, -cy, -cz)  
P' = M * P