>>9514944I imagine the object in Galilean space (fixed Euclidean structure) and imagine static rotation and arbitrary spacial shift s given by:
r' = Rr + Vt + s , s in Real^3
along with a constant temporal translation j given by:
t' = t + j , j in Real
where R stands for the 3 x 3 orthogonal rotation matrix given by:
R =
( cosa cosb cosa - sinc sina , cosc cosb sina + sinc cosa , cosc sinb)
( sinc cosb cosa - cosc sina , sinc cosb sina + cosc cosa , sinc sinb)
( sinb cosa , sinb sina , cosb )
Where a , b , c are 3 Euclidean angles of rotation. The domains of these angles are 0<=a<=2pi , 0<=b<=pi , and 0<=c<=2pi and the rotation matrix R satisfies |R| = 1 and RR^T = R^TR = I .
These generalized Galilean transformations are a set of linear equations. Given an inertial frame S, it can be carried over to another inertial frame S' in 10 possible ways corresponding to 3 spatial translations, 3 rotations, 1 temporal translation, and 3 for boosts by the constant velocity V constituting a 10-parameter Galilean group.
Representing an element of the group as h(R,V,s,j) the composition rule is defined as h3(R3,w,s3,j3) = h2(R2,v,s2,j2) o h1(R1,u,s1,j1)
where R3 = R2R1 , w = v + R2u , s3 = s2 + R2s1 + vj1 , j3 = j1 + j2 .
