Study On Two Kinds Of Rational Curves And Surfaces

In this paper,it's mainly about a new basis function found in triangle polynomial space and mixed trigonometric polynomial space.Two kinds of rational curve and surface's construction method and properties is introduced.And surface interpolation approximation problem in CAGD is studied.In addition,the paper also provide another shape preserving interpolation curve and its properties.Mainly researches and results are as follows:Firstly,it mainly narrates about the background and significance of rational spline interpolation with parameters and shape preserving interpolation curve in triangle polynomial space and algebraic-trigonometric polynomial space.Meanwhile it also introduces the relevant concepts of this thesis and its organizational structure.Secondly,Explicit expressions for the rational quadratic triangular Bézier curve with two shape parameters is introduced.From the study of properties of the curve,we found out that the same geometric characteristics with the traditional rational three times the Bézier curve: endpoint properties,symmetry,convex hull,geometrical invariability,variation reduction,etc.The example shows that the curve can not only present arc and ellipticarc accurately,but can also approach to the control polygon than three times rational Bézier curve.Approximation effect is better.Furthermore,this paper studies the smooth curve splicing.Under certain conditions,adjacent segments trigonometric polynomial curvecan can reach GC11,and GC22,continuous,so that it can provide an effective method for the design of free curve surface.In the end,it introduces the rational quadratic triangular Bézier surface with two shape parameters.Thirdly,based on the mixed trigonometric polynomial and algebraic polynomial space,a new rational cubic hybrid Hermite spline curve is constructed.The new constructed curve can change its shape from different parameters according to the designers' demand.Compared with the fixed Ferguson curve shape,rational cubic algebraic triangle hybrid Hermite interpolation curve has better practicability and better "soft" and approximation relative to the cubic spline curve.On the other hand,the structure of the spline curve inherited many excellent characters from trigonometric polynomial curve.Finally,This part introduces the rational last three algebraic triangular mixed Hermite surface.Fourthly,the paper introduces the shape preserving interpolation problem of cubic algebraic triangle hybrid Bézier spline curve.Sufficient and necessary conditions for the cubic algebra mixed Bézier spline curve shape preserving interpolation were derived from positive,monotonicity and convex array.Based on the control parameters of inequality,it can flexibly select parameters to achieve the effect of conformal.And the effectiveness and accuracy of this method are validated by examplesFinally,the main content of the thesis is summarized,and puts forward problems to be solved.