Geodesics Between 3D Closed Curves Using Path-Straightening

In order to analyze shapes of continuous curves in R, we parameterize them by arc-length and represent them as curves on a unit two-sphere. We identify the subset 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 shapepreserving transformations such as rotation and re-parametrization. To construct a geodesic in this shape space, we construct the shortest geodesic between the all possible transformations of the two end shapes; this is accomplished using an iterative procedure. We provide step-by-step descriptions of all the procedures, and demonstrate them with simple examples.