【作者】 郑雪芳；

【导师】 林意；

【作者基本信息】 江南大学 ， 计算机软件与理论， 2008， 硕士

【摘要】 在计算机辅助几何设计中,曲面上的插值曲线研究是一个十分重要的课题。本文主要讨论可展曲面上的插值曲线,经过分析,构造曲面上插值曲线的关键在于找到一个好的参数化方法,也就是如何将R~2上一条曲线对应到曲面上,使得到的曲面上曲线的扭曲尽可能小。基于这样的一种思想,结合微分几何理论,本文通过构造等距对应将可展曲面展成平面,从而将可展曲面上的曲线插值归结为通常的R~2上插值曲线的构造。分别针对已知型值点和型值点处的单位切矢以及只知道型值点而型值点处切矢未知的情况构造了C~1和G~2连续的插值曲线,并以可展曲面的三种类型:柱面、锥面、切线曲面为例构造了其上的插值曲线。文章最后还提出了一种选点修改法与能量法相结合的算法光顺可展曲面上的插值曲线。本文主要包括如下内容:1.对曲线曲面造型方法的相关内容作了简要介绍,简述了曲面上插值曲线的研究现状,回顾了曲线光顺技术的意义和发展现状。2.介绍了曲线曲面的基本理论,包括曲线的切矢、曲率、挠率以及曲线论基本公式,曲面论基本公式及曲面的法曲率、测地曲率、等距对应等。3.在已知型值点以及型值点处切矢的条件下,提出一种构造可展曲面上C~1连续的插值曲线的方法,采用积累弦长作为参数,无需反算控制顶点。4.在已知型值点而型值点处切矢未知的情况下,采用积累弦长作为参数,利用二次三角B-样条来构造插值曲线,无需反算控制顶点,并证明了所得插值曲线是G2连续的。5.提出一种选点修改法与能量法相结合的算法光顺可展曲面上的插值曲线,以圆柱面为例对其上的插值曲线进行了光顺处理,并推导了光顺以后控制顶点的坐标,用matlab软件绘出光顺后的曲线及曲率图,图例显示该算法具有满意的效果。更多还原

【Abstract】 In CAGD, the research on the interpolation curves on surface is a very important topic. This text is mainly about the interpolation curves on developable surfaces. After analyzed, the key of constructing interpolation curves on surfaces is to find a good parameterization method, namely how to map a curve of R~2 onto the surface, to minimize the distortion of the obtained curve on the surface. Base on such an idea, according to the basic principles of differential geometry, it constructs a isometric correspondence, and then, the developable surface will be developed into plane. Therefore, the interpolation curve on the developable surface can be ended into the construction of interpolation curve in R~2 generally. This paper put forward methods for constructing interpolation curves of C~1 and G~2 continuous, according to two situations, getting the points on surface with specified tangent vector, or just getting the points but no tangent vector. The three types of developable surface: cylndrical surface, conical surface and tangent surface ,are used as examples to construct the interpolation curves. And it also presents a new method mixing bad points selection with energy minimization, for fairing the interpolation curves on the developable surface.The content of this paper is mainly as following:1. Simply introduce the relevant content of curve & surface shaping method, briefly describe the present condition of interpolation curves on surface, and review the meaning and development of curves fairing.2. Introduce the basic theory of curve & surface, including the tangent vector、curvature、torsion of curve and the basic expressions of curve theory & surface theory, normal curvature、geodesic curvature、isometric correspondence of surface etc.3.When the points on surface are given with specified tangent vector, a method of constructing C1 continuous interpolation curves on the developable surface is presented. Take the accumulated chord length as parameter, without calculating the control points.4. Under the situation of just getting the points but no tangent vector, it takes the accumulated chord length as parameter,and uses the quadratic trigonometric B spline curve to construct the interpolation curve. Without calculating the control points. And it proves that the interpolation curve on the developable surface is G~2 continuous..5. A new method mixing bad points selection with energy minimization, for fairing the interpolation curves on the developable surface is presented. It takes the cylindrical surface as an example, to fair its upper curve. Deduce the coordinates of control points after fairing. Draw the curve and its curvature after fairing with the software MATLAB, and it shows that this algorithm is effective.更多还原