  We investigate the geometric characterization of pure state bipartite
entanglement of $(2\times{D})$- and $(3\times{D})$-dimensional composite
quantum systems. To this aim, we analyze the relationship between states and
their images under the action of particular classes of local unitary
operations. We find that invariance of states under the action of single-qubit
and single-qutrit transformations is a necessary and sufficient condition for
separability. We demonstrate that in the $(2\times{D})$-dimensional case the
von Neumann entropy of entanglement is a monotonic function of the minimum
squared Euclidean distance between states and their images over the set of
single qubit unitary transformations. Moreover, both in the $(2\times{D})$- and
in the $(3\times{D})$-dimensional cases the minimum squared Euclidean distance
exactly coincides with the linear entropy (and thus as well with the tangle
measure of entanglement in the $(2\times{D})$-dimensional case). These results
provide a geometric characterization of entanglement measures originally
established in informational frameworks. Consequences and applications of the
formalism to quantum critical phenomena in spin systems are discussed.
