Towards flattenable mesh surfaces

In many industries, products are constructed by assembled surface patches in ℜ³, where each patch is expected to have an isometric map to a corresponding region in ℜ². The widely investigated developable surfaces in differential geometry show this property. However, the method to model a piecewise-linear surface with this characteristic is still under research. To distinguish from the continuous developable surface, we name them as flattenable mesh surfaces since a polygonal mesh has the isometric mapping property if it can be flattened into a two-dimensional sheet without stretching. In this paper, a novel flattenable mesh surface (Flattenable Laplacian mesh) is introduced and the relevant modelling tool is formulated. Moreover, for a given triangular mesh which is almost flattenable, a local perturbation approach is developed to improve its flattenability. The interference between the meshes under process and their nearby objects has been prevented in this local flattenable perturbation. Both the computations of Flattenable Laplacian meshes and the flattenable perturbation are based on the constrained optimization technology.