Computing a Metric on the Shape Space of Tori

This paper explores shape comparison for genus-one surfaces, namely tori, through an intrinsic metric on their shape space. While metric structures for 2-spheres are well established, extending them to tori is substantially harder, due to their nontrivial topology and the absence of conformal diffeomorphisms in general. Drawing on Teichmüller theory, we formulate an algorithm that minimizes a distortion energy over diffeomorphisms within a canonical homotopy class to compute a metric on the shape space of Riemannian tori. This produces both a pairwise distance and an optimal, landmark-free correspondence. On the computational side, we discretize this energy on triangulated tori and optimize it directly in a pair of fundamental domains. A key theoretical contribution is a convergence theorem showing that the discrete energy converges to the smooth metric energy under mesh refinement. Experiments on flat tori, tori of revolution, and real-world scanned surfaces show that the method captures natural geometric variation and provides a robust tool for genus-one shape comparison.