Based On Harmonic Maps Theory Study Of The Surface Of Cnc Engraving

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.