Motivated by the emergence of rough surfaces in various areas of design, we address the computational design of triangle meshes with controlled roughness. Our focus lies on small levels of roughness. There, roughness or smoothness mainly arises through the local positioning of the mesh edges and faces with respect to the curvature behavior of the reference surface. The analysis of this interaction between curvature and roughness is simplified by a 2D dual diagram and its generation within so-called isotropic geometry, which may be seen as a structure-preserving simplification of Euclidean geometry. Isotropic dihedral angles of the mesh are close to the Euclidean angles and appear as Euclidean edge lengths in the dual diagram, which also serves as a tool for visualization and interactive local design. We present a computational framework that includes appearance-aware remeshing, optimization-based automatic roughening, and control of dihedral angles.

The property of a surface being developable can be expressed in different equivalent ways, by vanishing Gauss curvature, or by the existence of isometric mappings to planar domains. Computational contributions to this topic range from special parametrizations to discrete-isometric mappings. However, so far a local criterion expressing developability of general quad meshes has been lacking. In this paper, we propose a new and efficient discrete developability criterion that is applied to quad meshes equipped with vertex weights, and which is motivated by a well-known characterization in differential geometry, namely a rank-deficient second fundamental form. We assign contact elements to the faces of meshes and ruling vectors to the edges, which in combination yield a developability condition per face. Using standard optimization procedures, we are able to perform interactive design and developable lofting. The meshes we employ are combinatorially regular quad meshes with isolated singularities but are otherwise not required to follow any special curves on a developable surface. They are thus easily embedded into a design workflow involving standard operations like remeshing, trimming, and merging operations. An important feature is that we can directly derive a watertight, rational bi-quadratic spline surface from our meshes. Remarkably, it occurs as the limit of weighted Doo-Sabin subdivision, which acts in an interpolatory manner on contact elements.

We solve the task of representing free forms by an arrangement of panels that are manufacturable by precise isometric bending of surfaces made from a small number of molds. In fact we manage to solve the paneling task with surfaces of constant Gaussian curvature alone. This includes the case of developable surfaces which exhibit zero curvature. Our computations are based on an existing discrete model of isometric mappings between surfaces which for this occasion has been refined to obtain higher numerical accuracy. Further topics are interesting connections of the paneling problem with the geometry of Killing vector fields, designing and actuating isometries, curved folding in the double-curved case, and quad meshes with rigid faces that are nevertheless flexible.