【作者】 刘长明；

【导师】 檀结庆；

【作者基本信息】 合肥工业大学 ， 计算数学， 2004， 硕士

【摘要】 调整和改变曲线的形状是几何造型领域中常见的问题，本文重点讨论参数可调曲线的定义与推广，得到下述一些结果。 扩展了二次均匀B样条基函数，构造出三次和四次带局部参数λ_i的调配函数，推广后得到了n次的调配函数。它们具有二次均匀B样条基函数的性质，且用它们生成的分段多项式曲线具有与分段二次均匀B样条曲线相同的结构和几何性质。但与二次均匀B样条曲线相比，它们还有其自身的优点：首先，曲线的形状都可用参数λ_i进行局部调整：其次，四次调配函数所构造的曲线就可达到G~2连续；另外，为了满足实际应用中对曲线连续性的不同要求，可使用相应次数的调配函数来构造曲线。作为均匀B样条曲线的进一步扩展，作者对三次和四次B样条基函数进行扩展，构造了三B五次、四B五次、四B六次调配函数，从而产生了连续性分别达到C~3和C~4连续的多项式曲线，它们的形状都可以用参数λ进行调整。 对二次非均匀B样条作了进一步扩展，提高了曲线的连续性；曲线的每一段上都有一个局部控制参数，利用它们可以更有效的控制曲线的形状；同时，利用曲线的重节点可以很方便的在曲线上构造尖点。 作为B样条扩展曲线的应用，作者将上面构造的各次调配函数应用到三次α-B样条插值曲线上，得到下述结果。 利用三B四次调配函数对三次α-B样条插值曲线进行了扩展，扩展后得到的四次插值曲线在保留了三次α-B样条插值曲线的结构和性质的同时，增加了一个形状调节参数λ，从而扩大了参数对曲线的调节范围，使曲线更易于控制。作为三次α-B样条插值曲线的进一步扩展，也为了提高插值曲线的连续性，作者定义了相应次数的奇异调配函数，同时利用三B五次、三B六次调配函数分别构造了五次和六次参数可调的插值曲线，它们分别是C~3和C~4连续的。更多还原

【Abstract】 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.更多还原