# A Smoothness Energy without Boundary Distortion for Curved Surfaces

## TL;DR

This paper introduces a new generalized Hessian energy for curved surfaces that balances natural interior behavior with minimal boundary distortion, improving upon existing quadratic smoothness energies.

## Contribution

It proposes a novel Hessian energy formulation that accounts for intrinsic curvature and boundary behavior, discretized with Crouzeix-Raviart finite elements.

## Key findings

- Energy minimizers solve the Laplace-Beltrami biharmonic equation
- Discretization converges in experiments
- Reduces boundary distortion while maintaining natural interior behavior

## Abstract

Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions, or they do not correctly account for intrinsic curvature, which leads to unnatural-looking behavior away from the boundary. This leads to an unfortunate trade-off: one can either have natural behavior in the interior, or a distortion-free result at the boundary, but not both. We introduce a generalized Hessian energy for curved surfaces, expressed in terms of the covariant one-form Dirichlet energy, the Gaussian curvature, and the exterior derivative. Energy minimizers solve the Laplace-Beltrami biharmonic equation, correctly accounting for intrinsic curvature, leading to natural-looking isolines. On the boundary, minimizers are as-linear-as-possible, which reduces the distortion of isolines at the boundary. We discretize the covariant one-form Dirichlet energy using Crouzeix-Raviart finite elements, arriving at a discrete formulation of the Hessian energy for applications on curved surfaces. We observe convergence of the discretization in our experiments.