A Path-straightening Method for Finding Geodesics on Shapes Spaces of Closed Curves in R

In order to analyze shapes of continuous curves in R3, we parameterize them by arclength 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.