Developable surfaces are those that can be made by smoothly bending flat pieces without stretching or shearing. We introduce a definition of developability for triangle meshes which exactly captures two key properties of smooth developable surfaces, namely flattenability and presence of straight ruling lines. This definition provides a starting point for algorithms in developable surface modeling---we consider a variational approach that drives a given mesh toward developable pieces separated by regular seam curves. Computation amounts to gradient descent on an energy with support in the vertex star, without the need to explicitly cluster patches or identify seams. We briefly explore applications to developable design and manufacturing.

Noisy function on hemisphereLow-order boundary conditions Natural boundary conditions Fig. 1.Smoothing a noisy function with common low-order boundary conditions introduces a bias at the boundary: isolines exit perpendicularly. We propose using a different smoothness energy whose natural boundary conditions avoid this bias.In geometry processing, smoothness energies are commonly used to model scattered data interpolation, dense data denoising, and regularization during shape optimization. The squared Laplacian energy is a popular choice of energy and has a corresponding standard implementation: squaring the discrete Laplacian matrix. For compact domains, when values along the boundary are not known in advance, this construction bakes in low-order boundary conditions. This causes the geometric shape of the boundary to strongly bias the solution. For many applications, this is undesirable. Instead, we propose using the squared Frobenious norm of the Hessian as a smoothness energy. Unlike the squared Laplacian energy, this energy's natural boundary conditions (those that best minimize the energy) correspond to meaningful high-order boundary conditions. These boundary conditions model free boundaries where the shape of the boundary should not bias the solution locally. Our analysis begins in the smooth setting and concludes with discretizations using finite-differences on 2D grids or mixed finite elements for triangle meshes. We demonstrate the core behavior of the squared Hessian as a smoothness energy for various tasks.

Laplacian energy (zero Neumann boundary conditions) planar Hessian energyStein et al. [2018] our Hessian energy Fig. 1. Solving an interpolation problem on an airplane. Using the Laplacian energy with zero Neumann boundary conditions (left) distorts the result near the windows and the cockpit of the plane: the isolines bend so they can be perpendicular to the boundary. The planar Hessian energy of Stein et al. [2018] (center)is unaffected by the holes, but does not account for curvature correctly, leading to unnatural spacing of isolines at the front and back of the fuselage. Our Hessian energy (right) produces a natural-looking result with more regularly spread isolines, unaffected by the boundary.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.

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We present a simple and concise discretization of the covariant derivative vector Dirichlet energy for triangle meshes in 3D using Crouzeix-Raviart finite elements. The discretization is based on linear discontinuous Galerkin elements, and is simple to implement, without compromising on quality: there are two degrees of freedom for each mesh edge, and the sparse Dirichlet energy matrix can be constructed in a single pass over all triangles using a short formula that only depends on the edge lengths, reminiscent of the scalar cotangent Laplacian. Our vector Dirichlet energy discretization can be used in a variety of applications, such as the calculation of Killing fields, parallel transport of vectors, and smooth vector field design. Experiments suggest convergence and suitability for applications similar to other discretizations of the vector Dirichlet energy.