Research On The Properties Of Curves And Surfaces With Shape Parameters

We review the development history of parametric curves and surfaces and introduce briefly the studied results of the curves and surfaces with shape parameters.Then we research curves and surfaces with shape parameters. The main contributions are listed as follows:Firstly, Hyperbolic Bézier curve (briefly H-Bézier curve) with multiple shape parameters is presented in this thesis. The curve not only possesses most of the properties of the Bézier curve, but also can be adjusted totally or locally by taking the different values of shape parameters. With increasing of the shape parameters the curve can well approximate the control polygon. Moreover, the curve can represent hyperbolas and catenary exactly. At last, the C 1continuous joint of curve pieces is discussed and some examples illustrate the application in geometric modeling.Secondly, on the basis of study of the properties of H-Bézier curves, the H-Bézier surfaces are defined. The arbitrary subdivision algorithm along both u and v direction is presented, and the properties of subdivided surfaces are analyzed. G1 condition for H-Bezier surfaces in different directions is obtained, and simplified by choosing the control parameters properly.Thirdly, a class of 4-degree polynomial basis function with two shape parametersλ,μis presented, which is the extension of basis function of cubic Ball curve. Properties of this new basis are analyzed and the corresponding polynomial curve with two shape parameters is defined. This curve not only inherits the outstanding properties of cubic Ball curve, but also is adjustable in shape and fit close to the control polygon. And its shape is adjusted totally or locally through changing the value ofλ,μand approach to the given polygon from both sides. The C 1_continuity condition of two-piece of curves is discussed. Finally, the simulation example of the circle and the vase generated by the 4th order Ball curves with two shape parameters show the proposed method in this thesis is feasible and effective.