Extension Of B-spline And Its Applications

It is a familiar problem for changing shape of curves in geometric shape design, the article mainly focused on the definition and extension of parameterized curves which are adjustable, and the conclusion is described as follows.The quadratic uniform B-spline curves are extended, and a class of polynomial blending functions of degree 3 and degree 4 are presented in this paper, which can be extended to the case of degree n. They have the properties like the quadratic uniform B-spline basis functions. The piecewise polynomial curves generated by the above-mentioned functions possess the same structure and geometry properties as piecewise quadratic uniform B-spline curve. Comparing with the quadratic B-splinecurve, they have advantages by themselves: Firstly, the shape of the curves can be adjusted locally by the parameters i; Secondly, the curves formed by blendingfunctions of degree 4 can be G2 continuous. In addition, in order to meet various requests for continuity of curves in practical applications, corresponding polynomial functions can be used to construct the curves. As further extension of the uniform B-spline basis functions, the author extends the uniform B-spline basis functions of degree 3 and degree 4, and generates the blending functions of degree 5(3-B)n degree 5(4-B) and degree 6(4-B). As a result, the curves of C3 and C4 continuity can be generated, and the shape of the curves can be adjusted by the parameters X.The quadratic non-uniform B-spline curves are further extended and the continuity of curves is improved in this paper; With a local shape parameter in each piecewise curve, the shape of the curves can be controlled effectively; Moreover, cusps of curves can be generated conveniently on the curves while using multiple knots.As the application of the extension of the B-spline curves, the author applies the above polynomial functions to the cubic a -B-spline interpolation curves and gets the following results.Extending the cubic a -B-spline interpolation curves with the blending function of degree 4(3-B), we get the interpolation curves of degree 4. The curves have not only kept the structures and properties of the cubic a-B-spline interpolation curves but also increased a shape control parameter , which expands the adjusting ranges of the curves and make the curves easier to be controlled. As further expansion of the cubic a -B-spline interpolation curves, in order to raise the continuity of the curveseven, the author has defined the blending functions of corresponding degrees, constructed the adjustable interpolation curves of degree 5 and degree 6 with polynomial blending functions of degree 5(3-B) and degree 6(3-B) respectively. They are C3 and C4 continuous separately.