  In this article, we will formulate a mathematical framework that allows us to
treat character animations as points on infinite dimensional Hilbert manifolds.
Constructing geodesic paths between animations on those manifolds allows us to
derive a distance function to measure similarities of different motions. This
approach is derived from the field of geometric shape analysis, where such
formalisms have been used to facilitate object recognition tasks.
  Analogously to the idea of shape spaces, we construct motion spaces
consisting of equivalence classes of animations under reparametrizations.
Especially cyclic motions can be represented elegantly in this framework.
  We demonstrate the suitability of this approach in multiple applications in
the field of computer animation. First, we show how visual artifacts in cyclic
animations can be removed by applying a computationally efficient manifold
projection method. We next highlight how geodesic paths can be used to
calculate interpolations between various animations in a computationally stable
way. Finally, we show how the same mathematical framework can be used to
perform cluster analysis on large motion capture databases, which can be used
for or as part of motion retrieval problems.
