This is the summary of the 2-dim'l case.
Firstly, DDG of triangles 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.
Shown 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 2-dim'l shapes is proposed.
In the method, the shape of a trajectory is encoded into a {+1, -1}-valued sequence. The encoding rule is, change sign if the gradient changes.
Shown below are examples. These two trajectories on the left are encoded into the binary sequence on the right respectively.