Local PIA Of Triangular Bézier And Rational Triangular Bézier Surfaces

Reverse engineering plays a crucial role in CAGD, interpolation and approximation(fitting) problems of scattered data points are important issues in reverse engineering, and progressive iterative approximation(PIA) is an important method of the interpolation and approximation problems of scattered data points. By adjusting the control points of a surface iteratively, a series of surfaces are generated. When the number of iteration approach to infinite, if the limit of the surface interpolates initial data point, we say that the surface hold the PIA property. Because curves or patches are generated by progressive iterative approximation with well-adapted and finer precision, their roles pervade in various industries. So, many scholars research PIA of curves and surfaces recently. Based on local PIA of blending curves and patches,lexicographic order and Bernstein operator, this paper propose local PIA of triangular Bézier and rational triangular Bézier surfaces. Take advantage of knowledge of matrix theory, in terms of iterative format for adjusting vector and the relationship between eigenvalues of Bernstein operator and eigenvalues of corresponding collocation matrix, we get two methods of the convergence for triangular Bézier and rational triangular Bézier surfaces. Furthermore, applications to such local PIA in(rational)quadratic Bézier surface,(rational)cubic Bézier surface, parabolic in aspects are gave,the local PIA with well-adapted and finer precision can be illustrated by these applications. In the meanwhile, local PIA for surfaces have geometrical significance on large scale data fitting, adaptive data fitting, symmetric surface fitting. The major work is as follows:Chapter one is introduction. Firstly, it sketches the development of blending curves and patches. Secondly, a brief introduction of PIA of blending curves and surfaces researched at home and abroad. Lastly, it gives a brief introduction of the main contents of the thesis.Chapter two is preliminaries. It gives an introduction to the definition of lexicographic order, and the definition and major properties of triangular Bézier and rational triangular Bézier surfaces. Then it gives an introduction to the iterative process of whole PIA for Bézier curve. Finally it introduces the expression of Bernstein operator and lays a theoretical foundation for studies in following chapters.Chapter three makes use of lexicographic order, extend the local PIA property of the univariate basis of blending curves and patches to the bivariate Bernstein basisover a triangle domain. For the given data points, local PIA were obtained in terms of only adjusting control points of even permutations and only adjusting control points of odd permutations, respectively. And in the same time it proves the local PIA format has convergence when the iterative surface sequence interpolates the given data points.In the meanwhile, examples illustrate the local PIA of surfaces having finer precision on large scale data fitting.Chapter four is similar to chapter three, the expression of rational triangular Bézier surface transform to the expression of triangular Bézier surface, for the given data points, local PIA of rational triangular Bézier surface and relative examples are gave. Examples illustrate the local PIA of surfaces having finer precision on adaptive data fitting.Chapter five is based on literature [10], it gives prove process of range of eigenvalues for even permutations Bernstein operator equal to range of eigen-values for corresponding collocation matrix, and according to the spectrum radius between 0 and1, convergence of triangular Bézier and rational Bézier surfaces are obtained, application examples are gave. Examples illustrate the local PIA of surfaces having feasibility on symmetric surface fitting.Chapter six is the summary of this paper; it looks forward to the future work and puts forward problems for the further study.