Hyperbolic Hybrid Polynomial Curve And Its Properties

Bezier, B-spline, and their rational models play an important role in CAD/CAM systems. But these models show obvious shortcomings, such as no encompassing transcendent (i.e., nonalgebraic) curves, e.g. the circle, the cycloid, the catenary and the hyperbolic helix.This paper presents a new kind of curves which not only inherit good geometric properties of Bezier and B-spline models but also conquer some drawbacks of these models. We construct a new basis, to be called H-Bezier basis, of the space Kn =span{1,t,t2,…,tn-2,sht,cht} through mixing algebraic polynomial and hyperbolicfunctions. This new basis provides properties analogous to Berstein polynomials, including symmetry, zeros of the basis functions, positivity, normalization, etc. Based on this new basis, we define a new kind of curves, to be called H-Bezier curves, with control polygon. According to properties of H-Bezier basis, H-Bezier curves are endowed with wonderful geometric properties including interpolating at endpoints, convex hull, affine invariance and optimal shape preserving. Furthermore, a shape factor of the H-Bezier curve is introduced to control the shape of the designing curves. Hence designers can adjust the shape of curves by changing not only control points but also shape factor. Our experiments show that H-Bezier model approximate to the control polygon more closely than Bezier model. So they are suitable to shape design and modeling in CAD systems.Another task of this paper is to discuss Ball curves inKn. It is well known that Ball curves preserve geometric proerties of Bezier curves and can be degree-elevated and degree-reduced more rapidly. We find out the Ball basis of the space Kn and define the H-Ball curves with this Ball basis. H-Ball curves can be well applied for curves design and shape modeling in CAD systems and related fields in term of their geometric properties.This paper try to explore an alternative model of NURBS and bring forth a general method to construct basis functions which have geometric properties analogous to Bernstein functions.