What are the best point-to-point alignments between curves in space, whether closed or open? Given a collection of shapes, what is a representative mean? How can we characterize the primary modes of variation within a shape dataset?
Functional data analysis (FDA) is an area of statistical geometry that can answer these questions. Currently, I’m working on applying these analytical methods to various biological datasets to develop better, lower-dimensional characterizations of shape and to understand morphological diversification.
For more reading, a good source is Functional and Shape Data Analysis by Anuj Srivastava and Eric P. Klassen. A shorter overview of the central methodology is given in Registration of Functional Data Using Fisher-Rao Metric by Srivastava et al.