基于调和映射理论进行曲面数控雕刻的研究

【作者】 李洪涛；

【导师】 王晓明；

【作者基本信息】 大连理工大学 ， 机械设计及理论， 2004， 硕士

【摘要】 随着数控雕刻技术的发展，雕刻产品的精细度越来越高。但是进行曲面雕刻时存在一个棘手的难题，就是图案的扭曲变形。如何生成曲面雕刻的刀具运动轨迹来解决这个难题是打破曲面数控雕刻过程中的瓶颈的关键。 为了避免在复杂曲面的数控雕刻过程中出现图案的扭曲变形，本文提出了新的生成曲面数控雕刻刀具运动轨迹的方法。我们把应用平面数控雕刻技术生成的刀具运动轨迹，按照调和映射的对应关系映射到曲面上，生成曲面雕刻的刀具运动轨迹。由于调和映射具有等角变换的性质，所以本文提出的方法可以保证刻在曲面上图案基本不变形。 本文介绍了调和映射的基本理论，重点研究平面到曲面的调和映射。文中的曲面或用参数方程表达或用符号距离函数表达。对于这两种表达方式，分别应用变分法推导出平面到曲面的调和映射所必须满足的偏微分方程。这两个偏微分方程都是非线性调和方程。 直接求解非线性调和方程是很困难的，我们把非线性调和方程转化为非线性扩散方程，通过求解非线性扩散终值问题来得到调和映射。本文给出了求解非线性扩散终值问题的数值算法，并讨论了算法的稳定性，同时给出了编程实现算法的主要思路和重要程序的框图。 为了演示本文提出的生成曲面数控雕刻刀具运动轨迹的方法的有效性，我们进行了几个数值实验。文中分别依照初始映射、中间计算结果和调和映射的对应关系将图像贴到曲面上，并把贴图后的曲面透视投影到观察面上，从最开始的初始映射到最后的调和映射，透视效果逐渐改善，证明本文提出的方法是有效的。更多还原

【Abstract】 With the development of numerical control engraving technology, the precision of sculptures is better and better. But there is a thorny problem on sculpting on a complex surface. It is the pattern’s distortion. And it is the key to break the bottleneck to the process of numerical control sculpting on surfaces, that how to generate the tool-path of engraving on surfaces to solve the tough problem.To avoid the pattern being distorted in the process of numerical control sculpting on a complex surface, a new method is put forward to generate the tool-path of engraving on the surface. It maps the tool-path, which is generated by using the technology of sculpting on the plan, to the surface based on the harmonic mapping. Because the harmonic mapping has a conformal property, this method can preserve the pattern sculpted on surface to be unchanged.The basic theory of harmonic mapping is introduced, and the harmonic mapping from a plan to a surface is emphasized. In this paper, the surface is presented with a parameterized equation or a signed distance function. And to the both expression of the surface, two partial differential equations are achieved, which the harmonic mapping must satisfy, by using the variation approach. These partial differential equations are all nonlinear harmonic equations.It is difficult to solve these nonlinear harmonic equations directly, so these nonlinear harmonic equations are changed to nonlinear diffusion equations, and the harmonic mapping is obtained by solving the nonlinear diffusion equations in the stable state. The numerical algorithm for getting the solution of the nonlinear diffusion equations in the stable state is introduced, meanwhile the stability of the algorithm is discussed. The main idea for carrying out the algorithm is given, and the emphases program schemata are also presented.In order to show the effectiveness of the method, which is used to generate the tool-path of engraving on the surface, some numerical examples are given. The image is mapped to the surface with the initial mapping, the mappings of some computing steps and the harmonic mapping respectively, and then the surface is projected to the vision plan. These project effect are better and better from the initial mapping to the harmonic mapping, so the method is proved to be effective.更多还原