  In this paper we show that trajectory shape analysis should be performed only
after obtaining a proper representation before applying ordination methods. In
fact, studying the shape of a trajectory means studying how the deformation
changes along each path irrespectively of the actual shape to which these
deformations apply. The independence of the deformation from the shape to which
it is applied is critical: it implies that any shape variation between
individuals at the beginning of each trajectories must be completely filtered
out. A Parallel Transport, that can be based on various connection types, is
necessary to perform such kind of shape data centering. The Levi Civita
connection can also be used to transport a deformation. We demonstrate that
this procedure does not preserve deformation even in the ane case. We propose a
novel procedure called Direct Transport able to perfectly transport deformation
in the ane case and to better approximate non ane deformation in comparison to
existing tools.
