Research On Harmonic-Maps-based NC Surface Engraving

With the development of numerical control engraving technology, the problem of pattern's distortion has received more and more attention in sculpting on a complex surface, and many efforts have been made to try to solve this problem. This thesis focuses on how to generate the engraving tool-path on surfaces for solving this problem.A new method has been developed in this thesis, which can preserve the pattern sculpted on surface unchanged. Basically, the proposed method firstly generates the tool-path by using the technology of sculpting on plane, and then the tool-path is mapped to the surface based on the harmonic mapping theory.The basic theories of harmonic mapping and the harmonic mapping from a plane to a surface are emphatically discussed in this thesis. The target free form surface is constructed based on the cubic B-Spline surface, and a partial differential equation, which the harmonic mapping must satisfy, is achieved by using the variation approach. Because it is difficult to solve the harmonic equations directly, the harmonic equation has to be changed to diffusion equations, the solution of which helps to obtain the harmonic mapping. In the mean time, the algorithm for the solution of the diffusion equations is intensively discussed and the stability and the convergence of the algorithm are also proved.Some numerical examples are given in order to show the effectiveness of the method used to generate the engraving tool-path on the surface. The image is first mapped to the surface with the initial mapping and the harmonic mapping, and then the surface is projected to the vision plane. The melioration from the image mapping to the surface projection onto the vision plane proves the effectiveness of this method.