  Computing intrinsic distances on discrete surfaces is at the heart of many
minimization problems in geometry processing and beyond. Solving these problems
is extremely challenging as it demands the computation of on-surface distances
along with their derivatives. We present a novel approach for intrinsic
minimization of distance-based objectives defined on triangle meshes. Using a
variational formulation of shortest-path geodesics, we compute first and
second-order distance derivatives based on the implicit function theorem, thus
opening the door to efficient Newton-type minimization solvers. We demonstrate
our differentiable geodesic distance framework on a wide range of examples,
including geodesic networks and membranes on surfaces of arbitrary genus,
two-way coupling between hosting surface and embedded system, differentiable
geodesic Voronoi diagrams, and efficient computation of Karcher means on
complex shapes. Our analysis shows that second-order descent methods based on
our differentiable geodesics outperform existing first-order and quasi-Newton
methods by large margins.
