Research On Approximate Degree Reduction For Interval Rational Bezier Curves And Surfaces

Degree reductions of interval rational Bezier curves and surfaces are studied in this thesis. Based on the representation of interval rational Bezier curves and surfaces and by a serial of mathematical transformation, the degree reductions of them are converted to those of polynomials with upper bounds, then several algorithms are presented, with linear programming and optimal approximation methods. By relaxation of some constrained conditions, approximation effects of some of them are further improved.Firstly, some basic concepts and theories of interval algorithms and Bezier curves and surfaces are introduced so as to be used below.Secondly, as for degree reduction of interval rational Bezier curves, two methods are given: Pseudo Linear Programming Method (PLPM) and Pseudo Optimal Approximation Method (POAM). With the first method, multi-degree reduction can be executed at one time and certain continuity conditions can be satisfied; By relaxation of constrained conditions, great improvement on the LPM of literature [30] can be obtained when interval rational Bezier curves are degenerated to interval Bezier curves. Explicit computation formulas are presented when only one or two degrees reductions are required in the second method, and upper bounds for their errors are estimated. When compared with the first method, the second one is more effective.Thirdly, degree reduction of interval rational Bezier surfaces in rectangle domain is discussed. Two methods are given. One is obtained by using degree reduction of interval rational Bezier curves in two parameter directions according to the character of tensor product, which is called "Single Step Method". By comparison for degree reductions produced by the different order in two parameter directions, the relationships of the approximation results are discussed. The other, called "Integral Method", is got by converting the problem to degree reduction of bi-variable polynomials with upper bounds using approximation theory and the relationships between Chebyshev bases and Bernstein bases. The two algorithms are analyzed and their approximation results are compared.Finally, degree reduction of interval rational triangular B-B surfaces is discussed. The problem is also converted to degree reduction of bi-variable polynomials with upper bound. According to degenerating conditions of triangular B-B surfaces, an optimization model is constructed and optimal solutions under certain constrains are calculated. The constrained conditions are unproved using the tighter convex hull property of the control points degree elevated or subdivided, so the results of this method are better than those of literature [33].All algorithms above are accompanied with examples. Their results are analyzed and compared, which show the efficiency of the algorithms.