Developable Approximations For Discrete Surfaces

Nowadays,a new round of scientific and technological revolution and industrial transformation are advancing by leaps and bounds,which provides a historical opportunity for the intelligent development of manufacturing industry.In mathematics,a developable surface is a special ruled surface with zero Gaussian curvature everywhere,which can be formed by folding,bending or rolling the plane material without stretching or compressing,that is,without transverse deformation.Therefore,in the development of manufacturing industry in the past,developable surfaces have different applications in architecture,ships,airplanes,automobiles,clothing design and simulation,origami and plane parameterization.However,at present,the application of developable surfaces mainly appears in foreign research projects,and the research on developable surfaces in the past is mostly focused on manual design tools.Therefore,the intelligent application based on developable surfaces has a long way to go.This dissertation attempts to solve the problem of piecewise developable approximation,mainly focusing on how to automatically generate appropriate piecewise developable approximation results for a given input model.The work in this dissertation is divided into three parts,including a global developable approximation method and two different piecewise developable approximation methods.To generate the global developable approximation of the input mesh,researchers have proposed a variety of ways to define developability on discrete meshes.However,these definitions have some problems,for example,their results are affected by mesh connectivity,the generated seam curves are not salient,and the approximation error is large.Therefore,in the first work of this dissertation,a new edge-oriented discrete developability is proposed,and a global developable approximation method for deforming the input mesh based on this developability is given.For each edge,we propose the condition that the determinant of four matrices based on the edge normal of its neighborhood is zero,which is similar to the degenerated condition of Gaussian mapping in the continuous case.Theoretically,the edge-oriented developability includes the properties that the Gauss map degenerate,our developability definition is weaker than the definition based on hinge structure and little affected by mesh connectivity in special shapes.After that,we propose a deformation process for the input mesh to generate the global developable approximation based on the edge-oriented definition.In addition,in order to effectively optimize the deformation energy,we introduce different types of auxiliary variables and use the block nonlinear Gauss-Seidel method.For different models,the deformation process based on the above developability definition can get the results that are little affected by the mesh connectivity,have salient seam curves and small approximation error.Generally,the automatically generated piecewise developable approximation results for a given input mesh need to meet three requirements:(1)each patch is discrete developable,and flattened to a plane without any overlap;(2)the number of developable patches is small;and(3)there is high shape similarity between the piecewise developable output and the input.It is very challenging to achieve a favorable trade-off between the number of patches and the shape similarity,while the previous technology left room for reducing the number of patches and improving the shape similarity.In the second work of this dissertation,a new deformation-driven method is proposed to automatically generate the piecewise developable approximation with small number of developable surface patches and small approximation errors.The core of this method is to use an nearly developable mesh,which makes the segmentation stage simple and reliable.In order to reduce the difficulty of segmentation,we first use the deformation algorithm based on the edge-oriented developability to get the deformed results.Then,under the guidance of the seam curves of the deformed mesh,we propose a coarse-tofine segmentation technique to get the segmented mesh.Finally,we use the refinement process to get a proper balance between discrete developability and approximate error.Although this method can’t guarantee to find the best configuration of the developable patches,it shows fewer patches and smaller approximation error than the previous methods in practice.At the same time,the feasibility and effectiveness of the tool we developed are proved by applying it to various shapes,and we have made a physical model with paper to verify our results.In addition to the two requirements of the number of patches and approximation error,(1)the patch boundary is short,and(2)small patches and narrow regions should be eliminated in the piecewise developable approximation,which can further reduce our cost in the manufacturing process.These mutually restrictive requirements and the existence of discrete variables and discontinuous functions make it difficult to optimize these objectives at the same time in practice.The past work has left room for improvement in reducing approximation error,patch number,boundary length and narrow regions.Therefore,in the third work of this dissertation,a method based on evolutionary genetic algorithm is proposed to adapt to combinatorial and discontinuous targets to generate a suitable piecewise developable approximation.The fitness function including four quality measures and three genetic operations,namely selection,crossover and mutation,are designed.Then,we get the optimal segmentation by evolving a group of over-segmented initial populations with small approximation errors.In practice,in order to compute the approximation error in the fitness function efficiently,we use conformal mapping technology to transform the requirement of small approximation error into the requirement of low mapping distortion.Compared with previous methods,this method can generate piecewise developable approximation with smaller approximation error,shorter boundary length,fewer patches,fewer small patches and fewer narrow regions.The feasibility and effectiveness of the algorithm are also verified by different models.