Meschers: Geometry Processing of Impossible Objects

Summary

Fig. 1. The mescher is a geometry representation that allows rendering and relighting impossible objects (left), as well as performing intrinsic geometry processing operations like heat diffusion (center) and geodesic distance queries (right). Abstract Impossible objects, geometric constructions that humans can perceive but that cannot exist in real life, have been a topic of intrigue in visual arts, perception, and graphics, yet no satisfying computer representation of such objects exists. Previous work embeds impossible objects in 3D, cutting them or twisting/bending them in the depth axis. Cutting an impossible object changes its local geometry at the cut, which can hamper downstream graphics applications, such as smoothing, while bending makes it difficult to relight the object. Both of these can invalidate geometry operations, such as distance computation. As an alternative, we introduce meschers, meshes capable of representing impossible constructions akin to those found in M.C. Escher's woodcuts. Our representation has a theoretical foundation in discrete exterior calculus and supports the use-cases above, as we demonstrate in a number of example applications. Moreover, because we can do discrete geometry processing on our representation, we can inverse-render impossible objects. We also compare our representation to cut and bend representations of impossible objects. The Anatomy of A Mescher Impossible objects are integrable only locally but not globally. In plain English, this means that if we cover a corner of an impossible object with a sheet of paper, the visible part of it begins to look possible! Building on this insight, we construct a specialized mesh representation for Escheresque constructions. Unlike a typical mesh, which stores per-vertex 3D positions, a mescher stores per-vertex 2D screen-space positions and a per-edge depth difference. Whereas differences in depth across edges must sum up to zero as we travel around a standard mesh, this is not...