دانلود رایگان مقاله تغییر شکل free-form آنامورفیک

Abstract

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.

5. Conclusions

We have developed a simple, yet powerful method to create a digital anamorphosis, called AFFD (Anamorphic Free-Form Deformation), by endowing a rational FFD with an anamorphic character. The key idea is the use of homogeneous coordinates with origin at the viewpoint where the observer is located. In this setting, the problem becomes trivial, because to displace a certain point anamorphically (along a radial direction passing through the viewpoint), we only alter its last homogeneous coordinate. Instead of displacing an object point by point, we can displace in the same manner the NURBS control points of an object defined by NURBS curves or patches. In the case of a complex geometry, the AFFD provides a more convenient alternative to generate an anamorphic version. We embed the (planar or 3D) object in a (planar or 3D) structure and then deform the structure by applying an anamorphic transformation to the control points of the lattice. The user moves the control points along radial lines through the viewpoint to the desired final position, whereas the weights, inversely proportional to the distance to the viewpoint, come as a byproduct. Remarkable cases are the AFFD of a planar object with a rational linear Bézier triangle (resulting in a projection into another plane) or with a rational quadratic Bézier surface (corresponding to the inverse of a stereographic projection from a quadric). In case of a rational linear Bézier tetrahedron as deformation tool, the volumetric AFFD is nothing else than a perspective collineation, defined geometrically in a Bézier fashion. The AFFD gives a notable example where the rational model provides a clear advantage over its polynomial counterpart, as anticipated in any topic related to realistic perspective maps (Farin, 1989). Notwithstanding that the rational form introduces an unnecessary complication in certain constructions (Piegl et al., 2014), in the AFFD it is fully justified. Although the mathematical underpinnings involve homogeneous coordinates and projective space, the user input amounts to displacing affine points in Euclidean space. Thus, we avoid altogether a nuisance of the rational model pointed out by Farin (1992), namely having to choose the weights.