This equation for the trajectory of P can be inverted to compute the coordinate vector p in M as: p = [ T ( t ) ] − − 1 P ( t ) = [ p 1 ] = [ A ( t ) T − − A ( t ) T d ( t ) 0 1 ] [ P ( t ) 1 ] . This expression uses the fact that the transpose of a rotation matrix is also its inverse, that is: [ A ( t ) ] T [ A ( t ) ] = I .