Spirals and Thier Applications in CAGD

几何建模在各种物体的构造、设计和制造中起着重要的作用。除了在汽车和飞机制造、造船和一般产品设计等传统领域的重要性外，几何建模方法在各种现代工业中也被证明是不可或缺的，包括计算机视觉、机器人、金属成像、可视化纺织、设计、绘画和其他媒体。本研究旨在研究螺旋线及其在几何造型中的应用。螺旋线有几个优点，既不包含拐点，奇点，也不曲率极值。在视觉上令人愉悦的曲线设计中，这种曲线对于现有曲线的延伸和现有曲线之间的过渡是有用的。这种三次螺旋由三次样条组成，即，仅在特定位置处具有曲率极值的曲率连续曲线对于诸如公路或铁路路线或移动的机器人的轨迹或数控切割机械的切割路径的设计的应用是期望的。在表面设计中，特别是在演化表面和扫掠表面的设计中，使用这种曲线的好处是控制不想要的平坦点和起伏。首先，我们得到了每个子区间上给定数据的三次螺线条件。其次，我们提出了一种三次样条逼近给定数据的算法，并推导出一种更容易计算的算法。第三，我们探讨了使用这样的螺旋段的CAD。

Geometric Modeling plays a significant role in the construction, design and manufacture of various objects. In addition to its critical importance in the traditional fields of automobile and aircraft manufacturing, shipbuilding, and general product design, geomtric modeling methoda have also proven to be indispensable in a variety of modern industries, includind copmputer vision, robotics, medal imaing, visualization textile, designing, painting, and other media. The research aims to study spirals and their applications to geometric modeling. Spirals have several advantages of containing neither inflection points, singularities nor curvature extrema. Such curves are useful for extension of an existing curve and transition between existing ones in the design of visually pleasing curves. Such cubic spirals composed of cubic splines, i.e., curvature continuous curves with curvature extrema only at specified locations are desirable for applications as the design of highway or railway routes or the trajectories of mobile robots or the cutting paths for numerically controlled cutting machinery. The benifit of using such curves in the design of surfaces, in particular surfaces of evolution and swept surfaces, is the control of unwanted flat spots and undulations. First we have obtained the cubic spiral condition for the given data on each subintervals. Secondly, we have proposed an algorithm for a cubic spline approximation of the given data and derived an easier to calculate algorithm. Thirdly, we have explored the use of such spiral segments for CAD.