This is the summary of the 3-dim'l case.
Firstly, DDG of tetrahedrons is proposed.
In the construction, base space B is given as the collection of all "flat tiles". And tangent bundle TB over B is identified with the cartesian product of monimials { ... } and B.
Show on the right is fiber of TB, where slant tiles are mapped on a flat tile by projection pi.
Secondly, as an application, encoding of 3-dim'l shapes is proposed.
In the method, the shape of a trajectory is obtained by folding a tetrahedron sequence and encoded into a {+1, -1}-valued sequence. The encoding rule is, change sign if the gradient changes.
Shown below are examples. These two trajectories are encoded into the binary sequence on the right respectively.