Research On The Multiple Products Theory Of B-splines And Polynomial Approximation Techniques Of Rational Curves And Surfaces

NURBS(Non-Uniform Rational B-Spline) curves and surfaces and rational triangular Bezier surfaces are common tools in geometric design and modelling systems. Therein the multiple products of B-splines, the MCV(Monotone Curvature Variation) condition and evaluation of the derivative bounds of NURBS curves and surfaces, and the approximation of rational triangular Bezier surfaces using polynomial triangular Bezier surfaces have become current research hotspots, since they directly relate to the shape control, rendering efficiency, validity of algorithm, data interchange and transmission of CAD (Computer Aided Design) systems. However, there have no breakthroughs in these areas heretofore. In this dissertation, we have made deeply researches on these topics and provided abundant and innovative results as follows.Firstly, we create the theory of translating the product into the sum of B-spline functions, which represents multiple products of B-splines as a linear combination of some suitable B-splines. Recurring to the discrete B-spline theory and religious fine-draw analyses of the transformation of splines' knot vector spaces, some formulae concerning the degree and knot vector, of the multiple products of B-spline functions, are presented. Then Marsden's identity is generalized and the representation of the coefficients, of all terms of the multiple products of B-splines, is provided. Consequently, we obtain the formula of translating the product into the sum of B-spline functions, which can be directly applied to software coding in system development. The foundation of the multiple products of B-splines improves the function of design systems, enriches the theory of NURBS curves and surfaces, and impels NURBS curves and surfaces more widely applications in CAD.Secondly, the MCV condition for NURBS curves is offered. A MCV discriminant for a uniform planar rational cubic B-spline curve, most frequently used in engineering, can be converted into a higher degree B-spline function, with the formula of translating three products into the sum of B-splines. Applying the property of positive unit resolution of B-splines, a MCV sufficient condition for the curve segment is obtained. The result is simple and applicable in curve design, especially in curve fair processing.Thirdly, the evaluation of the derivative bounds of NURBS curves and surfaces is presented. The representation of uniform planar rational B-spline curves' scaled hodographs and their derivative magnitude bounds are derived, by applying discrete B-spline theory, Dir function indicating the direction of the Cartesian vector between homogeneous points, and the formula of translating the product into the sum of B-spline functions. As an application of the result above, an upper bound of a parametric distance between any two points in a uniform planar rational B-spline curve is further presented. Then, derivative bounds of NURBS curves with more complicated knot vectors are deduced, with the sophistications of some identities and inequalities. Finally, we obtain some derivative bounds of NURBS surfaces, by looking a rational surface as a dynamic curve, the locus of a rational curve. Investigations on estimating the derivative bounds of NURBS surfaces, enhance the efficiency of different algorithms of NURBS surfaces, and fill internationally this work blank.Fourthly, we inaugurate a new algorithm of approximating rational triangular Bezier surfaces by polynomial triangular Bezier surfaces, which has a sententious expression and insures the convergence of approximation. The main result is that a polynomial triangular Bezier sequence approximates the original rational triangular Bezier surface as the elevated degree tends to infinity, with inequality skills and infinitesimal analysis techniques. Especially, the arbitrary given order derived vector of the sequence converges uniformly to that of the approximated rational triangular Bezier surface. The polynomial triangular surface is constructed as follows. Firstly, we elevate the degree of the approximated rational triangular Bezier surface. And then a polynomial triangular Bezier surface is produced, who has the same order and new control points of the degree-elevated rational surface. The approximation method commendably solves two shortcomings-fussy expressions and the uninsured convergence of the approximation-of Hybrid algorithms for rational polynomial curves and surfaces approximation. Therefore, The method has theoretical and practical value, and further improves the function of geometric design systems.