Construction of G^1 Continuous Composite Curves and Surfaces

In order to break through the limitation that all parts of the B-spline curve and surface must have exactly the same degree while retaining the automatic smoothness of B-spline method, and to have shape adjustability independent of the control points, this paper proposes a method for constructing G^1 continuous composite curves and surfaces. Firstly, a group of basis functions of degree n(n>=2) which contain two free parameters is constructed, and the properties are analyzed. Based on these basis functions, a new kind of degree n curves and surfaces which have the same structure as the Bézier curves and surfaces is defined. They contain the degree n Bézier curve and surface as special cases. Next, the G^1 smooth join conditions of the new curves and surfaces are analyzed. According to the join conditions, the piecewise combination curves and surfaces based on the new curves and surfaces are defined. They contain the quadratic uniform and quasi-uniform B-spline curve and surface as special cases. The combination idea is the same as the B-spline method, but the combination method is different. The definition automatically ensures the G^1 continuity of the composite curves and surfaces at the joints. The position of the end (corner) points and the internal shape of the composite curves and surfaces can be adjusted by changing the free parameters, and the adjustment can be both global and local. This method facilitates the design of complex curve and surface.