دانلود رایگان مقاله همگرایی مختصات گرانیگاهی به هسته گرانیگاهی

Abstract

We investigate the close correspondence between barycentric coordinates and barycentric kernels from the point of view of the limit process when finer and finer polygons converge to a smooth convex domain. We show that any barycentric kernel is the limit of a set of barycentric coordinates and prove that the convergence rate is quadratic. Our convergence analysis extends naturally to barycentric interpolants and mappings induced by barycentric coordinates and kernels. We verify our theoretical convergence results numerically on several examples.

6. Conclusion

We have shown that barycentric coordinates on polygons and barycentric kernels on smooth domains are closely linked by the limit process that involves denser and denser polygons converging to a domain with smooth boundary. The convergence rate of this limit process is quadratic and also applies to barycentric interpolants and barycentric mappings. Therefore, barycentric coordinates defined over polygons with only dozens of vertices could be used instead of computationally more demanding barycentric kernels. A detailed (non-asymptotic) analysis that would give bounds on the error between the smooth and discrete interpolants/mappings goes beyond the scope of the current paper. We plan to investigate it for certain classes of functions ci in (6) in the future. Our theoretical results were confirmed by several numerical experiments on barycentric coordinates and kernels as well as the interpolants and mappings that they induce. Since generalisations of barycentric coordinates (Floater et al., 2006) from 2D to 3D are readily available (Ju et al., 2007), we also plan to look in this direction.