Triangulated Surface Flattening: Research & Applications

Surface flattening is an important base for many technologies in the area of CAD&CG. Unfortunately, seldom complex surfaces can be flattened analytically. Benefited from the advance of 3D scanning and modeling, high complex surfaces can be described with triangles. Therefore we can flatten a surface numerically by means of flattening triangles. Although, in the last two decades, great progress has been made in the study of triangulated surface flattening, flattening a high complex surface remains an open problem even today.On the base of analysis of its history and the problems remained, we find out the factors influencing the result of surface flattening. They are the surface developability, mesh quality, method for surface flattening. Consequently, a strategy that is helpful to conquer those disadvantage factors is proposed: surface developability is improved by surface trimming. Surfaces are flattened with quasi-symmetrically distributed triangle strips based on a mass-spring model. High complex surfaces are flattened with hierarchy meshes based on bi-mapping. The integration of two surface flattening algorithms improves the ability on flattening high complex surface with low mesh quality from the crosswise and the longitudinal two aspects.Based on Gaussian curvature, a method for evaluating the developability of triangulated surfaces is presented. Surface developability provides guides for cutting surface reasonably and creating hierarchy mesh properly. Vertices and edges on the triangulated surface are abstracted into a mass-spring model which is used to control the shape of triangles. In order to more efficiently decrease the shape deformation of triangles during surface flattening and decrease the influence of mesh quality in the flattening results, two improved mass-spring models are presented respectively, they are the model with a crossed spring and the model with a constraint spring. With those two improved models, we can efficiently decrease the deformation energy in surface flattening results and decrease the influence of mesh quality in the flattening results.Two algorithms for triangulated surface trimming are proposed. They are the algorithm of interactive surface trimming with recognizing cutting region automatically and the algorithm of automatic surface trimming based on generating and optimizing cutting path automatically. In the interactive surface trimming algorithm, cutting lines are drawn interactively on model surfaces to outline the shape of 3D patches, then boundaries of each patch is searched out automatically by picking a point inside the patch, and finally the patch is trimmed out automatically with its boundaries. Several types of cutting lines with different trimming mechanisms are defined, which makes the design flexible. The automatic surface trimming algorithm regards decreasing Gaussian...