Research On The Extension Of Bézier Curve And Surface Modeling With Shape Parameters

Bezier curve and surface geometry has many good geometrical properties, which is widely used in computer-aided geometric design. It has also become an important research tool for curve and surface modeling design. Bezier curve is mainly used to describe and draw curve according to approximation theory, which represents rigidity. The curve is determined after the control polygon was determined, which brings the great difficult for the adjustments and modifications of the curve. In order to resolve this inflexibility of the engineering design caused by rigidity, this paper uses the introducing parameters method in Bernstei polynome, which can accomplish the local and global adjustment of the curve and surface. Except for that, this paper also implements a smooth stitching for the curves and surfaces. between the perfect description of a complex combination of curves and surfaces, which gives the perfect description of a complex combination for curves and surfaces.The paper generalizes the Bernstein Bases replacing the variable of t by the function f(t). This new curve that using this bases not only possesses all of the characters of Bézier curve according to choosing the function f(t),but also produces some advantages. for example, we can easily change the curve’s degree by adjusting factor; but u do not change the shape of the curve. Bezier curve and proposed the Bézier curve corresponds to the points on the curve is different, so this curve is more free and agile than Bezier curve when curve patches. this bases and curve have some applying and study value.one kind of curve with there shape parameters is defined, which are both extensions of Bézier curve and Bézier curve. The basis functions of the new curve have been discussed. The properties, application, the G1 and G2 continuity conditions of two- piece of the curves of the curve have been studied. Meanwhile, in the case of the same control polygon, by adjusting the value of the shape parameter can generate different approximate the control polygon curve. Its can also precisely represent or approach quadric curves through changing the value of the shape parameters. In addition,conic said parabola and leaf design examples are given, at the same time also gives a new curve and curve resulting from the joining together of rotator, this makes in the design of the curve in the free curve surface has high application value.Finally, the paper construct three generalized Bézier surface with shape parameters, which is based on the three generalized Bernstein function with shape parameters. The surface not only retains the original geometric properties of Bézier surface, but also this surface produced some new features. It studies and gives the feature of the new surface and special surface degradation structure. Meanwhile, in order to solve the problem that the single curve can’t express complex curves for the style design. This paper inferred the geometrical condition of smooth connection for the two cubic generalized Bézier surface which are adjacent. At the same time, it also gives the procedures of connection and example of geometric modeling.