过渡曲线的设计与应用-数学专业论文.docxVIP

万方数据 万方数据 摘 要 本文主题为一些关于过渡曲线的设计与应用的研究，主要涉及过 渡曲线的构造方法、曲率单调性的研究、端点处的连续性以及一些相 关的实际应用。因过渡曲线在 CAD、工业数控加工、道路线形设计和 机器人行走轨迹等领域都有非常广泛的应用，对其研究具有十分重要 的意义。 本文在绪论部分简要地介绍了一些关于过渡曲线的研究背景和研 究现状。第二章通过对基函数的研究构造了 6 条只有 3 个控制顶点的 类三次 Bézier 曲线，给出了曲率单调的充分条件。第三章分别以sin t 、 t 、1 和 cost 、t 、1 为基函数构造了两条都含形状参数且只有 3 个控制 顶点的??-曲线，也给出了曲率单调的充分条件。第四章从提高过渡曲 线连续性的角度构造了使其在端点处达到 C k 连续的最低次多项式势函 数和从调整过渡曲线形状的角度构造了使含形状参数的过渡曲线在端 点处保持 C1 连续的混合三角势函数。第五章分析了旋叶式压缩机气缸 型线中出现拐点的原因，提出了型线设计的新思路。从降低过渡曲线 次数的角度设计了吸气容积小而内容积比大的三次 Bézier 型线，适用 于小排气量高内压比的机器；从提高吸气容积的角度设计了吸气容积 大而内容积比小的类四次 Bézier 型线，适用于大排气量低内压比的机 器。第六章主要是总结本篇论文的工作并提出今后研究的方向。 关键词：过渡曲线，曲率单调，类三次 Bézier 曲线，??-曲线 I ABSTRACT The theme of this thesis is several kinds of designing and applications about transition curves, mainly related to the construction of the transition curves, the study of curvature monotony, the continuity at endpoints and some practical applications. Since transition curves have been widely used in fields of CAD, numerical control machining, road design and robot trajectories et al, the research of them is very important. In chapter1, the research background is given. In chapter 2, six quasi-cubic Bézier curves with only three control points are constructed by means of a study on base functions, and the sufficient conditions of monotone curvature are given. Also, we proposed two types of new ??- curves with a shape parameter in chapter 3, which only have three control points. One adopts sin t , t and 1 as its basis while another adopts cost , t and 1, also the sufficient conditions for monotone curvature of these ??- curves are deduced. Then, in chapter 4, a new kind of minimal degree polynomial potential function which can make transition curve reaching C k continuity at endpoints and another new kind of mixed trigonometric potential function with a shape parameter which can reach high coincidence at one side of transited curves on conditions of keeping C1 continuity at endpoints are constructed. The reasons for