基于高斯映射的可展曲面等曲率离散问题研究  

Gauss Map Based Developable Surface Discretization

作　　者：张利萍 彭彦平[2] ZHANG Liping;PENG Yanping(School of Engineering,Westlake University,Hangzhou 310024;Department of Mechanical Engineering,Dalian Polytechnic University,Dalian 116034)

机构地区：[1]西湖大学工学院,杭州310024 [2]大连工业大学机械工程与自动化学院,大连116034

出　　处：《机械工程学报》2020年第5期226-232,共7页Journal of Mechanical Engineering

摘　　要：提出可展曲面及其折展结构的空间高斯映射数学建模方法与分析理论,实现空间折展特性分析与折纸曲面的离散化研究,并继而可以进一步引申到可展曲面的等效数学模型的空间机构学研究。通过建立可展曲面的高斯映射方法,建立空间折展曲面与复杂折展结构的高斯球面曲线,实现三维空间折展结构降维到二维球面曲线,且以高斯映射方法建立可展折纸曲面的数学模型基础理论,并通过曲线曲面理论,将三维的可展曲面问题转化为高斯球面曲线的数学分析。继而通过折展曲面与折展单元的高斯球面曲线的对应研究分析,且由球面曲线的离散得到折纸曲面的近似多面体折展结构,并为其等效机构学特性研究奠定基础,进一步将曲面折展转化为平面折展可对应的现有机构和机器人学问题,实现曲面折展问题与平面折展问题对接,并探讨其工程应用依据。Gauss map of the continuous curved surfaces is lent to model the geometry properties and configuration information of spatial curved origami. The Gauss spherical curves achieve the goal of dimension deduction of the discretization problem in which R3 surface discretization is shifted to spherical curve discretization. Then, the equal segments are utilized to discretize the spherical curves first. The resulting points on the spherical curves present the discretized normal directions of the curved surface. Thus, the discretized ruling lines, as new folds in the approximate polyhedron, can be obtained. The aim of this work has been to provide a curvature discretization for developable surfaces based on the Gauss spherical curves. The directed foldable units are identified and the discrete operations of these units are proposed. More complex curved origami can be analyzed with its decomposition and the constructive units. It provides a differential geometry means to deal with the instantaneous folding movements. Folding with non-flat curvature is exploited. The equal curvature discretization of curved surface is realized.

关 键 词：曲面折展 可展曲面 高斯映射 曲面离散 曲率 

分 类 号：TG156[金属学及工艺—热处理]