Research On The Properties Of Curves In The Non-Algebraic Polynomial Space

In order to overcome the shortcomings of curves and surfaces modeling in algebraic polynomial space, many scholars propose other forms of curves and surfaces in the non-algebraic polynomial space. Based on the study of the scholars, this thesis does some study as follows:Firstly, through analysis of the properties of quartic C-curves, we present an approach of constructing planar piecewise quartic C-Bézier curves and quartic C-B spline curves with all edges tangent to a given control polygon. The C-Bézier curve segments are joined together with C 1continuity and the Quartic C-B spline closed curves and open curves are C 3 continuous. All curves are shape preserving to their tangent polygons. All control points of the curve segments can be calculated simply by the vertices of the given tangent polygon. Finally some numerical examples illustrate that the method given in this paper is effective.Secondly, based on the analysis of the properties of cubic H-Bézier curves, a subdivision algorithm is proposed, to compute the control parameters and control points of the two subcurves subdivided by any point of cubic H-Bézier curves. The connection conditions between cubic H-Bézier curves and cubic Bézier curves are derived and the applications of cubic H-Bézier curves in the surface modeling are given. The obtained results, which are simple and intuitionistic, can effectively improve the shape representation and control of cubic H-Bézier curves.Thirdly, an effective subdivision formula for H-Bézier curves of degree five is presented. Furthermore, it is proved that the control polygons generated by the subdivision converge to the original H-Bézier curves of degree five. Two important properties, the variation diminishing (V-D) property and convexity preserving property, are proved for H-Bézier curves of degree five.