For two curves on a surface, one interesting question is how to quantify how similar the curves are, motivated by applications in GIS data analysis, medical imaging, and computer graphics. Geometric measures such as Hausdorff and Frechet distance are computable, but often not desirable since they do force the deformation to move continuously in the ambient space.

In this talk, we'll consider measures that instead are based on homotopy or homology. Such deformations will generally look to minimize some quantity associated with the homotopy or homology, such as total area swept or longest intermediate curve, for example. We will survey several measures which have been introduced and studied in recent years, giving structural properties as well as considering the complexity of the problem or known algorithms to compute it.