Modeling Research On Quartic Rational Bézier Curve And Surface

Computer aided geometric design(CAGD) mainly researches curves and surfaces which vary in the free and complex forms, that is to say, curves and surfaces of freedom forms. Bézier curves and surfaces are widely applied in computer graphics and have many good properties in shape design of industrial products.In recent years, Bézier curves and surfaces have extremely important function in the applications of engineering design, such as air planes, ships and so on. Firstly a curve shape is obtained by designing a curve that crosses some data points. Then the corresponding shape surface is formed by interpolation on these curve shapes. So Bézier curves and surfaces show their strong vitality in practice and receive wide attentions.Firstly, the definition of rational quartic Bézier curve is introduced and its properties are described. The shape of rational quartic Bézier curve is modified by changing the weights corresponding to control points instead of control points themselves. The G2 continuity for the joining between two adjacent rational quartic Bézier curves is realized. Based on this the G2continuity for the joining among three adjacent quartic rational Bézier curves are achieved further.Secondly, the parameterization of rational quartic Bézier curve is discussed. The expression formula of rational quartic Bézier curve is deduced by barycentric coordinates. The parameter and inner weight factors are calculated inversely by given five control points and a point on the rational quartic Bézier curve within the convex hull formed by all the control points of the curve. The equivalent property between the weight transformation for spatial rational quartic Bézier curve and parametric transformation of the curve under the condition of keeping the shape unchanged is investigated.Altering the control points or weights of rational quartic Bézier curve is one of important approaches in achieving the shape modification. In practical applications, the modification with predestinate target is more hoped to be realized, for example, to make the modified curve pass a given point. The method to make the modification is given both by modifying the control points and by modifying weight factors.According to the theory of Bézier surface, by taking the property that the curve and its tangent plane is continuous on the common boundary, the joining between adjacent rational Bezier surface is studied and the G1 continuity conditions of two adjacent rational Bézier surfaces of double four degrees are given, therefore more adjustable shape parameters can be obtained.