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Current quadratic smoothness energies for curved surfaces either exhibitdistortions near the boundary due to zero Neumann boundary conditions, or theydo not correctly account for intrinsic curvature, which leads tounnatural-looking behavior away from the boundary. This leads to an unfortunatetrade-off: one can either have natural behavior in the interior, or adistortion-free result at the boundary, but not both. We introduce ageneralized Hessian energy for curved surfaces, expressed in terms of thecovariant one-form Dirichlet energy, the Gaussian curvature, and the exteriorderivative. Energy minimizers solve the Laplace-Beltrami biharmonic equation,correctly accounting for intrinsic curvature, leading to natural-lookingisolines. On the boundary, minimizers are as-linear-as-possible, which reducesthe distortion of isolines at the boundary. We discretize the covariantone-form Dirichlet energy using Crouzeix-Raviart finite elements, arriving at adiscrete formulation of the Hessian energy for applications on curved surfaces.We observe convergence of the discretization in our experiments.

A Smoothness Energy without Boundary Distortion for Curved Surfaces