We propose an efficient representation for studying shapes of
closed curves in R^n. This paper combines the strengths of two
important ideas - elastic shape metric and path-straightening
methods - and results in a very fast algorithm for finding
geodesics in shape spaces. The elastic metric allows for optimal
matching of features between the two curves while
path-straightening ensures that the algorithm results in geodesic
paths. For the novel representation proposed here, the elastic
metric becomes the simple L^2 metric, in contrast to the past
usage where more complex forms were used. We present the
step-by-step algorithms for computing geodesics and demonstrate
them with 2-D as well as 3-D examples.