In order to analyze shapes of continuous curves in $\real^3$, we
parameterize them by arc-length and represent them as curves on a
unit two-sphere. We identify the set denoting the closed curves,
and study its differential geometry. To compute geodesics between
any two such curves, we connect them with an arbitrary path, and
then iteratively straighten this path using the gradient of an
energy associated with this path. The limiting path of this
path-straightening approach is a geodesic. Next, we consider the
shape space of these curves by removing shape-preserving
transformations such as rotation and re-parametrization. To
construct a geodesic in this shape space, we seek the shortest
geodesic between the all possible transformations of the two end
shapes; this is accomplished using an iterative minimization. We
provide step-by-step descriptions of all the procedures, and
demonstrate them with examples.