دانلود رایگان مقاله ساخت RAGS از طریق اسپلاین همگن

Abstract

Recently, a construction of spline spaces suitable for representing smooth parametric surfaces of arbitrary topological genus and arbitrary order of continuity has been proposed. These splines, called RAGS (rational geometric splines), are a direct generalization of bivariate polynomial splines on planar triangulations. In this paper we discuss how to construct parametric splines associated with the three homogeneous geometries (spherical, affine, and hyperbolic) and we also consider a number of related computational issues. We then show how homogeneous splines can be used to obtain RAGS. As examples of RAGS surfaces we consider direct analogs of the Powell–Sabin macro-elements and also spline surfaces of higher degrees and higher orders of continuity obtained by minimizing an energy functional.

6. Numerical experiments

The following numerical examples illustrate RAGS surfaces obtained by methods described in previous sections. The surfaces displayed in Figs. 7 and 8 are examples of local and global interpolation by C1 Powell–Sabin macro-elements for the mesh in Fig. 6 (left). In the former case, the required directional derivatives are estimated by the local approach presented in Section 4.1, which guarantees that the RAGS spline has the quadratic reproduction property. For the surface in Fig. 8, the derivatives are computed by requiring that the surface minimize the energy functional (13). As a consequence,the latter approach is global in nature and, as can be reasonably expected, it results in better aesthetic quality. Finally, the example in Fig. 9 is generated by the global interpolation method described in Section 4.2, employing degree-6 patches and C2-continuity conditions. It turns out that each surface is tangent to its respective control net at the vertices of , in the same way as in the case of standard splines. The illustrations in Figs. 7, 8, and 9 depict, from left to right, collections of individual surface patches, the final surface plots, the Gaussian curvature, and the zebra-line plots.