  Shape spaces are fundamental in a variety of applications including image
registration, morphing, matching, interpolation, and shape optimization. In
this work, we consider two-dimensional shapes represented by triangular meshes
of a given connectivity. We show that the collection of admissible
configurations representable by such meshes form a smooth manifold. For this
manifold of planar triangular meshes we propose a geodesically complete
Riemannian metric. It is a distinguishing feature of this metric that it
preserves the mesh connectivity and prevents the mesh from degrading along
geodesic curves. We detail a symplectic numerical integrator for the geodesic
equation in its Hamiltonian formulation. Numerical experiments show that the
proposed metric keeps the cell aspect ratios bounded away from zero and thus
avoids mesh degradation along arbitrarily long geodesic curves.
