Surface modeling and flattening for products fabricated by slightly-extensible planar materials

In those industries whose products are designed in 3D but fabricated by planar materials, a challenge work is to find out a 2D pattern for a given 3D design, and the 2D pattern should be warped back to the 3D shape with slight extension. Constraints coming from industries like length control on feature curves and boundary interpolation are needed to be enforced. To solve the aforementioned problems, we have proposed several approaches.Length-aware surface flattening is very useful for generating 2D patterns made of slightly-extensible materials. WireWarping method presented by Wang, 2008 is exploited to generate 2D patterns with invariant lengths of feature and boundary curves. However, strict length constraints on all feature curves sometimes cause large distortions on 2D patterns, especially for those 3D surfaces which are highly non-developable. Then, we present a flexible and robust extension of WireWarping by introducing a new type of feature curves named elastic feature, which brings flexibility to shape control of the resultant 2D patterns. On these new feature curves, instead of strictly preserving the exact lengths, only the ranges of their lengths are controlled. To achieve this function, a multi-loop shape control optimization framework is proposed to find out the optimized 2D shape among all possible flattening results with different length variations on those elastic feature curves, while the lengths of other feature curves are kept unchanged. Besides, we also present a topology processing algorithm on the network of feature curves to eliminate cases that lead to numerical singularity. The new proposed method is named as WireWarping++. Experimental results show that the WireWarping++ can successfully flatten surface patches into 2D patterns with more flexible shape control and more robust numerical performance.As an alternative to surface flattening, flattenable surface processing approaches try to process an input model into a flattenable surface where flattenable surface is a polygonal mesh surface that can be unfolded into a planar patch without stretching any polygon. Prior approaches result in either a flattenable surface that could be quite different from the input shape or a (discrete) developable surface has relative simple shape.To overcome the aforementioned shortages, our first attempt is a local flattenable processing approach. In stead of processing the input model by a global optimization, the local approach adjusts the positions of vertices one by one via a controllable Laplacian evolution. The computation speed of local approach is quite fast so that we develop an interactive tool based on it. The interactive tool can improve the flattenability of the processed model efficiently meanwhile having better shape approximation compared with the result obtained by FL-mesh processing proposed by Wang, 2007.To achieve a flattenable surface with good shape approximation to the input model, we also proposed a new method for computing a slightly stretched flattenable mesh surface M from a piecewise-linear surface patch P in 3D, where the shape approximation error between M and P is minimized and the strain of stretching on M is controlled. Firstly, we introduce a new surface modeling method to conduct a sequence of nearly isometric deformations to morph a flattenable mesh surface to a new shape which has a better approximation of the input surface. Secondly, in order to get better initial surfaces for fitting and overcome topological obstacles, a shape perturbation scheme is investigated to obtain the optimal surface fitting result. Lastly, to improve the scalability of our optimal surface fitting algorithm, a coarse-to-fine fitting framework is exploited so that very dense flattenable mesh surfaces can be modeled and boundaries of the input surfaces can be interpolated.Besides modeling onproducts fabricated by slightly-extensible materials, we also try to extend our work to modeling on compression garment. A calibration based method is proposed consist two aspects: 1) a material testing for establishing the relationship between length change and compression; 2) a length control flattening. Some preliminary results are presented.