The Analysis Of Spatial Model In Developable Surface Origami

Origami has the characteristics of simplicity,low cost and strong plasticity,which is a better material of model compared with other materials.In addition,the developable surface is a kind of special ruled surface,and its excellent properties are applied in many aspects.Therefore,it is significant to research the developable surfaces in origami.Firstly,the developable surface is selected as the research object,and its definition is given and the parameter equation is obtained by analysis.In addition,the viewpoint that discreting the developable surface with discreteing the directrix in developable surface is put forward.The curve whose curvature of directrix is a constant is discreted with the method of equal arc length.If the curvature of directrix is always vary,including the plane curve and space curve,the curve is discreted with the method of constant curvature variation.Then,error analysis of the two discreteness meothod is given.In the error analysis,five different discrete scales are used for the arc on the interval,and then the lesser discrete scale is received by comparing the MATLAB mapping.For the different curvature of plane curve in the interval,the errors of ?k=0.1 and ?k=0.2 are calculated and compared,and ?k=0.1 is gained as the discrete standard.For the space curve,the errors of ?k=0.1 and ?k=0.05 are calculated and compared,and ?k=0.05 is gained as the discrete standard.Secondly,the discrete idea of Gauss map is proposed.The spherical curve of a single developable surface(cylinder,cone and tangent surface)is gained by the isometric mapping relation of the Gauss map.And I have received the spherical curves of the two developable surfaces,including cylinder and cylinder,cylinder and cone,cone and cone.Among them,according to the different connection modes,each of the two developable surfaces unit can get three kinds of spherical curve.Furthermore,I have reached a conclusion that the rule of the spherical curve of multiple developable surfaces.The next step is to discrete the obtained spherical curve.For a single developable surface,I adopt the method of discreteness with equal spherical curve arc length.When two developable surfaces share a curve crease,I use the method of discreteness with equal quantity.At last,the concept of folding angle is given and the convexity and concavity of the adjacent developable surface are analyzed and summarized.And I also present a folding model of multiple conical surfaces,which is composed of conic curve(parabola,circle)and straight line.Then,different postures are gained by different folding angles.Next,I select one of the folded states in order to discrete with the principle of Gauss map and then use the MATLAB to make its spherical curve.After discretization,selecting a part of the multiple cone folding model to establish the rotation vector model.