Using isometries for computational design and fabrication

Abstract

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.

Dr. KID: Direct Remeshing and K-set Isometric Decomposition for Scalable Physicalization of Organic Shapes

“…One of its main uses is generating isometric segments of the input surface. This complex geometric problem enhances reusability, reducing both complexity and costs [19,22,26,50]. Isometric decomposition has applications in tiling [16], modeling, and fabrication [26].…”

“…Isometric patterns are highly encouraged in architectural geometry and civil engineering. In this regard, Jiang et al [22] used isometric bending of surfaces via a small set of molds to create manufacturable tiles, which addresses the problem of representing free forms. They only used constant Gaussian curvature for this paneling.…”

Discrete Isothermic Nets Based on Checkerboard Patterns

“…A new discretization, suggested by [13], is based on the checkerboard pattern inscribed in a quadrilateral net constructed by connecting edge midpoints. This approach has already proven to be useful in various applications [6,12,7]. Its effectiveness suggests that there is more to the concept than just the good numerical approximation qualities already hinted at by [13].…”