Equivalent Geometric System Analysis Of Developable Surface Origami

A developed surface is a kind of surface that can be transformed into a plane.There is can spread along the curve crease so that developed into a plane.Consequently,we regard the origami model as developable surface flexagon.The expanding and folding of the flexagon model is reversible,what is model,the folding model of the 3D space can trans-formed into a plane through expanding.Vice versa,by folding the designed two-dimensional figure along the curve crease,the designers can obtain the 3D space that they eager to have.On the basic of mathematical geometry,this paper explores and analyzes the mathematical design of complex and developable surface,and puts forward two-dimensional graphics with complex designed developed surface by means of the combination of developed surface folding unit.Firstly,according to the type of crease curve,we explore the problems of connection between two basic moveable surfaces.Simultaneously,on the basis of the crease curve,we designed developable unit according to the curve type of the crease and analyze it.The unit is the basis of the designing of developable surface's two-dimensional graphics so that we can establish the related developable surface unit database,which can provide reasonable resource and information for the design of complex and developable surface.Secondly,through the combination and analysis of the surface unit,we can discuss the problem of connection of any developable surface.At the same time,through the analysis of these surface units,we summarize the two-dimensional combination of all the developed surface elements.For the purpose of exploring the principles of combination,we analyze the reason of the uncomfortable combination,and take measures to turn.Thirdly,By the means of the combination and connection of the basic developable surface,relying on unit database,and the its assembley drawings,we make many design examples of multi-conical surface,surgical clamp and multi-layer basic curve combination,So that we can explore and summarize the principles of combination of the complex developed surface's two-dimensional graphics.Finally,with the help of the mathematical differential geometry,the building of Frenet frame on the 3D foldable crease model,the calculating of the folding angle of the space model,all of these can help us build up the mapping relation of the 2D surface plane and the 3D space model,then,giving us a mathematical condition that a 3D space model can be developed into a flat.