Parameterization for fitting triangular mesh

In recent years, with the development of 3D data acquisition equipments, the study on reverse engineering becomes more and more important. However, the existing methods for parameterization are hard to ensure that the parametric domain is rectangular, and the parametric curve grid is regular. In order to overcome the limitations, we present a novel method for parameterization of triangular mesh in this paper. The basic idea is two-fold: first, because the isotherms in the steady temperature do not intersect with each other, and distribute uniformly, it guarantees that there is no singularity (fold-over) in the parameterization; second, a 3D harmonic equation is solved by the finite element method to obtain the steady temperature field on a 2D triangular mesh surface with four boundaries. Therefore, our proposed method avoids the embarrassment that it is impossible to solve the 2D quasi-harmonic equation on the 2D triangular mesh without the parametric values at mesh vertices. Furthermore, the isotherms on the temperature field are taken as a set of iso-parametric curves on the triangular mesh surface. The other set of iso-parametric curves can be obtained by connecting the points with the same chord-length on the isotherms sequentially. The so obtained parametric curve grid is regular, distributes uniformly, and can map the triangular mesh surface to the unit square domain with boundaries of mesh surface to boundaries of parametric domain. It ensures that the triangular mesh surface or point cloud can be fitted with the NURBS surface.