Study And Application Of C-Bézier Curves And Surfaces Of Degree N

This paper ,from the theory and application viewpoint ,profoundly studies theC-Bézier curves of degree n and surfaces of degree n× m and pays attention to theproperties as the high order derivative , the degree reduction ,composition of C-Bézier curves and surfaces and etc, and investigates the occurrence of inflection points,singularities on C-Bézier curves of degree 3 as well. The research achievements and main contents are as follows: 1.The derivatives for C-Bézier curves of degree n are calculated and animportant property is obtained: when α → 0+ ,the k- order derivative for the C-Bézier curves of degree n equals that of the Bézier curves of degree n. The formulae ofthe curvature of C-Bézier curves, the Gaussian and mean curvature of C-Béziersurface are derived and expressed with simple geometric quantities got from thecontrol mesh. It is easier and more geometric intuition to calculate than those fromdifferential geometry. 2.Two methods for degree reduction of C-Bézier curves are presented. One formultidegree reduction of C-Bézier curves is developed by using both the property fordegree elevation of C-Bézier curves and the theory of generalized inverse matrix. Theother for degree reduction of C-Bézier curves is put forward based on the degeneratecondition of C-Bézier curves and the constrain optimization and an error of thedegree reduction is also estimated. Further the scheme is combined with a subdivisionalgorithm to generate lower degree curves with lower error. Geometric continuitybetween piecewise segments is also considered in the degree reduction process .Thetwo methods are compared. Finally, the first method for degree reduction isgeneralized to the case of C-Bézier surfaces. 4. At first the other method is developed to prove connection Matrix for highorder geometric continuity of C-Bézier curves. then from an algebraic angle anecessary and sufficient condition with G2 continuity between piecewise C-Béziersegments is obtained and from a geometric angle the composite C-Bézier curves ofdegree 3 and 4 with G1 continuity and G2 continuity is producedrespectively .Moreover, it is proved that G1 3-degree and G2 4-degree C-Béziercurves can become C1 and C 2 UAT B-spline curves by taking certain parametersrespectively. Finally The composite C-Bézier surfaces with G1 continuity is alsodiscussed. 5. The convexity, the cusp, the loop and the inflection points on the planar cubicC-Bézier curve on the plane of λμ are determined. A necessary and sufficientcondition for the cubic C-Bézier curve being torsion curve is gotten. At the same timethe property that there are no cusp, no loop and no generalized inflection on suchcurves are shown, which are very useful when using C-Bézier curves for geometricmodelling. Finally, the degree reduction and pieced together C-Bézier curves and surfacesabove are compared with those on Bézier curves and surfaces. The programs of theabove methods and algorithms are developed, which shall be used in the system ofsurface modeling for CAD/CAM.