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.

In an optical anamorphosis, an object is seen distorted unless the viewer is positioned at a specific point, where the object appears normal. We describe how to endow a rational Free-Form Deformation with an anamorphic character in a simple manner, obtaining an AFFD (Anamorphic Free-Form Deformation). Given a (planar or 3D) initial object, which will appear normal from the desired viewpoint, we deform the object with a rational Bézier surface or volume as deformation tool. To achieve the desired deformation, the user input amounts to displacing the control points of the deformation tool along radial directions through the viewpoint, whereas the weights come as a byproduct. Mathematically, the deformation means changing the last homogeneous coordinate of the control points. An AFFD defined by a linear Bézier tetrahedron can be regarded as a user-friendly way to construct a perspective collineation. In this case, or when the deformation tool is a Bézier triangle of degree one, the AFFD transforms NURBS to NURBS keeping the original degree. With a deformation tool of higher degree, the rational composition required to obtain the exact result yields NURBS also of higher degree. For a quadratic Bézier triangle as deformation tool, our AFFD coincides with the inverse of a stereographic projection.