  In the elastic shape analysis approach to shape matching and object
classification, plane curves are represented as points in an
infinite-dimensional Riemannian manifold, wherein shape dissimilarity is
measured by geodesic distance. A remarkable result of Younes, Michor, Shah and
Mumford says that the space of closed planar shapes, endowed with a natural
metric, is isometric to an infinite-dimensional Grassmann manifold via the
so-called square root transform. This result facilitates efficient shape
comparison by virtue of explicit descriptions of Grassmannian geodesics. In
this paper, we extend this shape analysis framework to treat shapes of framed
space curves. By considering framed curves, we are able to generalize the
square root transform by using quaternionic arithmetic and properties of the
Hopf fibration. Under our coordinate transformation, the space of closed framed
curves corresponds to an infinite-dimensional complex Grassmannian. This allows
us to describe geodesics in framed curve space explicitly. We are also able to
produce explicit geodesics between closed, unframed space curves by studying
the action of the loop group of the circle on the Grassmann manifold. Averages
of collections of plane and space curves are computed via a novel algorithm
utilizing flag means.
