Many algorithms on meshes require the minimization of composite objectives, i.e., energies that are compositions of simpler parts. Canonical examples include mesh parameterization and deformation. We propose a second order optimization approach that exploits this composite structure to efficiently converge to a local minimum.Our main observation is that a convex-concave decomposition of the energy constituents is simple and readily available in many cases of practical relevance in graphics. We utilize such convex-concave decompositions to define a tight convex majorizer of the energy, which we employ as a convex second order approximation of the objective function. In contrast to existing approaches that largely use only local convexification, our method is able to take advantage of a more global view on the energy landscape. Our experiments on triangular meshes demonstrate that our approach outperforms the state of the art on standard problems in geometry processing, and potentially provide a unified framework for developing efficient geometric optimization algorithms.

This paper presents an algorithm for morphing between closed, planar piecewise‐C1 curves. The morph is guaranteed to be a regular homotopy, meaning that pinching will not occur in the intermediate curves. The algorithm is based on a novel convex characterization of the space of regular closed curves and a suitable symmetric length‐deviation energy. The intermediate curves constructed by the morphing algorithm are guaranteed to be regular due to the convexity and feasibility of the problem. We show that our method compares favorably with standard curve morphing techniques, and that these methods sometimes fail to produce a regular homotopy, and as a result produce an undesirable morph. We explore several applications and extensions of our approach, including morphing networks of curves with simple connectivity, morphing of curves with different turning numbers with minimal pinching, convex combination of several curves, and homotopic morphing of b‐spline curves via their control polygon.