Mesh Surface Flattening And Developability Optimization

Developable surfaces are widely used in industrial applications due to their nice capability of being made from a flat sheet without any stretching and tearing. Hence, developable surfaces have attained a lot of research interests in the field of computer aided geometric design and computer graphics. With the development of 3D acquisition technologies and hardware devices, discrete polygonal mesh surfaces have become increasingly popular. It is worthwhile to study the discrete counterpart of developable surface in digital geometry processing. In this paper, we investigate the discrete developable surface and its associated problems. The main contributions of this paper are as follows.In general, only developable surface can be flattened into plane without any distortion. When flattening a non-developable surface, our goal is to minimize the geometric distortion. We present a novel parameterization method for a non-closed triangular mesh. For every flattened 1-ring neighbors, the local geometry structure is represented as local parametric coordinates. Then the global optimal parametric coordinates are attained by aligning all the local parametric planes. The boundary conditions are not necessary in our method. In addition, our method can operate directly on mesh surface which has holes without any preprocessing of surface partition or hole filling. Linear constraints are allowed in the parameterization in a least squares sense. Our method is very suitable for computer graphics applications which require parameterization with low geometric distortion, such as texture mapping.Planer parameterization method assumes that the mesh surface is a topological disc, or has inner boundaries. For closed mesh, we have to segment it into patches, and then parameterize each patch into plane. We propose an interactive mesh cutting tool, which is intuitive and easy-to-use. Based on sketching interface, users can segment the mesh instantly by simply drawing freehand sketches on the mesh which mark the seed region for each patch. And the cutting boundary of each patch is computed and refined automatically. The segmentation algorithm is based on an improved region growing algorithm, which is fast and provides instant visual feedback to the users. Combining segmentation, planer parameterization and texture synthesis techniques, we have a rich toolbox for texture mapping.Developable surface can be identified by computing its Gauss curvature, which is identically zero. For non-developable surfaces, there is no unified formula to quantify how far they are from being developable. We present a new method to evaluate the developability of the mesh surface. To improve the developability of a given mesh with minimum shape change, a energy function is designed to simultaneously ensure that the optimized mesh become more developable, fits the input data, and satisfies the smooth requirements of the application. A iterative numerical solution is proposed for solving this optimization problem. By using the nearly interpolation technique, the tolerance of the input mesh and the optimized mesh can also be controlled. Feature lines specified by the users can keep fixed during our optimization.Fascinating and elegant shapes may be folded from a single planar sheet of material, if one incorporates curved folds into the design. We investigate the continuous and discrete properties of the developable surfaces along curved folds, and present an optimization based computational framework for design and digital reconstruction of surfaces which can be produced by curved folding. Given a nearly developable triangular mesh, we detect its rulings, and construct a initial quad mesh using those rulings. An optimization based computational framework is proposed to refine the initial quad mesh. Based on the optimization framework, more applications are introduced, such as developable surface design with minimum bending energy, bending in the presence of a curved fold, etc. Our work not only contributes to applications in architecture and industrial design, but it also provides a new way to study the complex and largely unexplored phenomena arising in curved folding.