The Theories And Applications Of Algebraic-trigonometric-combined Curves

Among the modeling methods used in the CAD parametric curves and surfaces, the technique of blending curves and surfaces is being wildly study. This paper is arranged in the following two parts.In the first part of this paper, we will investigate the construction of the blended curves in the blended space Ω_m=span{1,t,t²,---,t^m-4,sint,cost,t sint,t cost}. Themainly reason to construct curves in the space Ω_m is that it can encompass more parametric curves, such as the Archimedean spiral, involutes of circles, the conical helix and so on. Thus, in the space Ω_m, we will construct AT-Bezier curves, uniformAT-spline curves, nonuniform AT-spline curves, which share the same properties of the Bezier curves, uniform B-spline curves, nonuniform B-spline curves, respectively. After repeatedly integrated the initiate bases, we get the bases of arbitrary order. After that, the blended curves are constructed using the bases and the control points. Consequently, we find some useful properties of these curves. For examples, the corner cutting polygon serial is convergent to the corresponding curve;The uniform AT-spline curve has a subdivided matrix. The nonuniform AT-spline curves can degenerated to NUAT or nonuniform B-spline.In the second part, we will use the C-Bezier and H-Bézier to derived a the tensor product representation of the general helicoid minimal surface. Firstly, for the helicoid patch, we get the control points and the geometric construction of the control mesh. Secondly, for a patch of the general helicoid minimal surface, we obtain its control points. Finally, since the helicoids and the catenoids are special general helicoid minimal surface, we get a transformation, which can transform a helicoid patch to a catenoid patch while the blended patches are also general helicoid minimal surface...