Metrics on shape space are used to describe deformations that take one shape
to another, and to determine a distance between them. We study a family of
metrics on the space of curves, that includes several recently proposed
metrics, for which the metrics are characterised by mappings into vector spaces
where geodesics can be easily computed. This family consists of Sobolev-type
Riemannian metrics of order one on the space Imm(S1,R2) of
parametrized plane curves and the quotient space Imm(S1,R2)/Diff(S1) of unparametrized curves. For the space of open
parametrized curves we find an explicit formula for the geodesic distance and
show that the sectional curvatures vanish on the space of parametrized and are
non-negative on the space of unparametrized open curves. For the metric, which
is induced by the "R-transform", we provide a numerical algorithm that computes
geodesics between unparameterised, closed curves, making use of a constrained
formulation that is implemented numerically using the RATTLE algorithm. We
illustrate the algorithm with some numerical tests that demonstrate it's
efficiency and robustness.Comment: 27 pages, 4 figures. Extended versio