I have a rigid 3D homogeneous transformation matrix 4*4 which includes Rotation (Roll, Yaw and pitch) plus the translation(X,Y and Z).I want to split this transformation into N small and equal transformations where: T Original = T . T . T . . . . . TIs it theoretically OK to just extract the angles and translation values from T Original and divide them by N then construct the T matrix from them? If not, how can I achieve that?

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I have a rigid 3D homogeneous transformation matrix 4*4 which includes Rotation (Roll, Yaw and pitch) plus the translation(X,Y and Z).I want to split this transformation into N small and equal transformations where:      T          Original        =  T  .  T  .  T  .  .  .  .  .  TIs it theoretically OK to just extract the angles and translation values from       T          Original       and divide them by   N then construct the   T matrix from them? If not, how can I achieve that?

Ueberbachge · Accepted Answer

Subdividing the rotation angle the way you propose works, but the translation is more complicated. W.l.o.g. suppose that   T is of the form      [                            R                                  t                                                            0            T                                    1                      ]    .Then      T    2    =      [                                        R            2                                            t          +          R                  t                                                            0            T                                    1                      ]          T    3    =      [                                        R            2                                            t          +          R                  t          +                      R            2                            t                                                            0            T                                    1                      ]      ⋮        T    n    =      [                                        R            n                                                (                          ∑                              k                =                0                                            n                −                1                                                    R              k                        )                            t                                                            0            T                                    1                      ]    .So,   t  =            (              ∑                  k          =          0                          n          −          1                            R        k            )              −      1              t      orig  , assuming that the matrix is nonsingular.Off the top of my head, I can’t think of a way to simplify this sum of powers of   R. Since   (  R  −  I  )      (          ∑              k        =        0                    n        −        1                    R      k        )    =      R    n    −  I,  it’s tempting to say that             (              ∑                  k          =          0                          n          −          1                            R        k            )              −      1        =  (      R    n    −  I      )          −      1        (  R  −  I  ), but every 3-D rotation has 1 as an eigenvalue and       R    n   is a rotation (the rotation part of the original matrix), so unfortunately       R    n    −  I is singular.

istupilo8k · Accepted Answer

Anyway you can find angle and divide by N or directly get it from the matrix by using formulas for sine cosine of       α    N

I have a rigid 3D homogeneous transformation matrix 4*4 which includes Rotation (Roll, Yaw and pitch

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