Poincare group

The Poincaré Group is 10 mathematical transformations that can be applied to different coordinates of the space-time graph (x, y, z, ct), but does not affect the "spacetime interval". The "spacetime interval" is roughly the 4-D equivalent of 3-D "distance".

These transformations are:

Spatial translations:
$T_{x}:(x,y,z,ct)=(x+/epsilon,y,z,ct)$
$T_{y}:(x,y,z,ct)=(x,y+/epsilon,z,ct)$
$T_{z}:(x,y,z,ct)=(x,y,z+/epsilon,ct)$

Time Translations:
$T_{t}:(x,y,z,ct)=(x,y,z,ct+/epsilon)$

Spatial Rotations:
$R_{x}:(x,y,z,ct)=(x,cos/thetay+sin/thetaz,-sin/thetay+cos/thetay,ct')$
$R_{y}:(x,y,z,ct)=(cos/thetax+sin/thetaz,y,-sin/thetax+cos/thetaz,ct')$
$R_{z}:(x,y,z,ct)=(cos/thetax+sin/thetay,-sin/thetax+sin/thetay,z,ct')$

Lorentzian Transformations:
$L_{x}:(x,y,z,ct)=(x',y,z,ct')$
$L_{y}:(x,y,z,ct)=(x,y',z,ct')$
$L_{z}:(x,y,z,ct)=(x,y,z',ct')$