How to make impossible objects possible: Anamorphic deformation of textured NURBS

Abstract

Most Escher-like impossible figures admit a realization as 3D objects, when seen from a specific viewpoint. To avoid the restrictions of polyhedral models, we explore how to build these 3D objects with curved surfaces, through the prism of CAGD. Such impossible objects can be regarded as optical anamorphisms, a visual illusion based on depth misperception, which can be achieved by using standard CAGD techniques, in particular homogeneous coordinates and NURBS. The main tool is an anamorphic deformation, i.e., along radial lines through the viewpoint. Deforming NURBS boils down to the straightforward exercise of displacing the control points along such radial directions and simultaneously changing their weights. Thus, we change the distance to the observer of different parts of an object, which leads to depth misperception and apparent contradiction. The deformation alters shading, but the introduction of textures in parameter space, unaffected by the deformation, deceives the eye and helps conceal this artifact, as 3D printouts corroborate. Examples confirm the superiority of our textured NURBS over simpler polyhedral models.