Meschers: Geometry Processing of Impossible Objects

2026-05-14

AI Summary

Existing methods struggle to accurately represent Escher-like impossible objects while simultaneously supporting standard geometric processing operations. This work proposes Meschers—a mesh representation based on discrete exterior calculus—that enables, for the first time, cut-free and bend-free conformal modeling of impossible objects. The formulation inherently supports a range of geometric processing tasks, including smoothing, relighting, and distance computation, and can be integrated with inverse rendering for appearance optimization. Experimental results demonstrate that Meschers significantly outperforms existing approaches, which rely on cuts or bends, while preserving visual impossibility. The method thus provides a unified and efficient framework for modeling and applying impossible objects.

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.

Problem

• impossible objects
• geometry processing
• computer representation
• discrete geometry
• Escher-like constructions

Innovation

• impossible objects
• discrete exterior calculus
• geometry processing
• inverse rendering
• Escher-like structures