Construction Of A Non-uniform Triviariate S₄～2 B-Spline

The spline function, which is piecewise smooth function, plays its important role inthe fields of aircraft manufacture, ship designing and it has become a unreplacableabletool in CAGD, CAD, CAM and so on. Meanwhile, it is one of basic methods for theapproximation or fitting of scatter data and it can be applied to the fields of wavelet andfinite element. On the other hand, as the development of the theory of multivariate splinefunctions, it is found that the spline function has various relationships to some basic areasof mathematics, such as abstract algebra, algebraic geometry, differential equations andso on.It is well known that spline functions have been widely applied in many areas, es-pecially with its application range greatly extended by its close relation with waveletanalysis. Since extensive study has been carried out on uni-and bivariate splines, manyresearch results on the generalization of wavelet basis can be found in[31]. However, ow-ing to some intrinsic difficulties in spline research in extending from 2D to higher dimen-sions, including 3D, the research on non-tensor product splines is quite limited. Althoughthere have been some reports on higher dimensional splines[27, 25, 31-41], such as sim-plex splines[25] and box splines[32], few results are suitable for direct application, whichgreatly restricts the applications of splines. Since the application of higher dimensionalsplines is required in many fields such as large-scale scientific computing, wavelet analy-sis and 3D visualization, the research on higher dimension splines is extremely necessary,especially the research on spline functions on non-uniform partition.The main work of this thesis is on the B-spline on the non-uniform partition.It includes five chapters. Chapter one summarizes the previous research on splinefunctions and introduces the theory of smooth cofactors by R. H. Wang and the B-splinemethod and the B-net method. Chapter two discusses bivariate spline functions on thenon-uniform type-2 triangulations. And we studied the trivariate box splines in chapterthree, especially the trivariate B-spline on non-uniform partition. Then, in chapter fourS₄² and S₃¹ B-splines in the R³ space are constructed via integration method-and type-3tetrahedron partition is introduced in this chapter. At last, we summarize the work wehave done and look forward to the future work in chapter five.