  In character animation, direction invariance is a desirable property. That
is, a pose facing north and the same pose facing south are considered the same;
a character that can walk to the north is expected to be able to walk to the
south in a similar style. To achieve such direction invariance, the current
practice is to remove the facing direction's rotation around the vertical axis
before further processing. Such a scheme, however, is not robust for rotational
behaviors in the sagittal plane. In search of a smooth scheme to achieve
direction invariance, we prove that in general a singularity free scheme does
not exist. We further connect the problem with the hairy ball theorem, which is
better-known to the graphics community. Due to the nonexistence of a
singularity free scheme, a general solution does not exist and we propose a
remedy by using a properly-chosen motion direction that can avoid singularities
for specific motions at hand. We perform comparative studies using two
deep-learning based methods, one builds kinematic motion representations and
the other learns physics-based controls. The results show that with our robust
direction invariant features, both methods can achieve better results in terms
of learning speed and/or final quality. We hope this paper can not only boost
performance for character animation methods, but also help related communities
currently not fully aware of the direction invariance problem to achieve more
robust results.
