Parametrization and computations in shape spaces with area and boundary invariants

Shape spaces play an important role in several applications in robotics, most notably by providing a manifold structure on which to perform motion planning, control, behavior discovery and related algorithmic operations. Many classical approaches to defining shape spaces are not well suited to the needs of robotics. In this abstract, we outline an approach to defining shape spaces that address the needs of such problems, which often involve constraints on area/volume, perimeter/boundary, etc. Using the simple example of the space of constant-area and constant-perimeter triangles, which are represented as Riemannian manifolds, we demonstrate efficient solutions to problems involving continuous shape evolution, optimal sampling, etc.