Research On Curves And Surfaces With Shape Parameter In CAD

The parameter curves and surfaces modeling are used very widely in CAD. In this paper we profoundly research a new technique in parameter curves and surfacesmodeling-parameter curves and surfaces modeling with shape parameter. Firstly, weintroduce the development history and background of parameter curves and surfaces modeling with shape parameter. Then basis of initial function and node vector, the different basic spline functions with shape parameter are constructed by recursion and integral approach. The properties of basic spline functions with shape parameter, the range of shape parameter and the VD property of spline curves with shape parameter are all discussed lastly. The main contributions are listed as follows:(1) Uniform Spline curves and surfaces with shape parameterChoosing three initial functions, then we can get k -th (k ≥ 2) order uniform B spline curves and surfaces with shape parameter, trigonometric polynomial and hyperbolic polynomial uniform B spline curves and surfaces with shape parameter. They all possess one adjustable shape parameter and can reach C^k-2 that is higher than βspline curves which is G² continuous. The different curves can be adjusted by changing shape parameter in invariable control polygon and at the same time they have the same structure and geometry properties as uniform B spline curves. With the elevation of order, feasible range of the shape parameter value is extended. The circle, ellipse and helices can be represented with trigonometric polynomial uniform B spline curves with shape parameter. The hyperbola can be represented with the hyperbolic polynomial uniform B spline curves with shape parameter.(2) Bezier Spline curves and surfaces with shape parameterChoosing three initial functions, then we can get k -th (k ≥ 2) order Bezier curves and surfaces with shape parameter, trigonometric polynomial and hyperbolic polynomial Bezier curves and surfaces with shape parameter. They all possess one adjustable shape parameter and have the same structure and geometry properties as Bezier curves. The different curves lying on the Bezier curve can be created by changing shape parameter in invariable control polygon and approximate to the control polygon as the increase of the...