Theory And Method Of Curved Origami Design Based On Gaussian Curve

At present,the research on the analysis of the connectable surface joints by means of differential geometry at home and abroad is still in its infancy.In 1976,American scholar David A.Huffman was the originator of the origami field.He proposed that the relative properties of the left and right curved shapes of the five basic curve creases are fixed.Based on this,we further study the curved properties on both sides of the crease curve.David A.Huffman first proposed the concept of Gaussian ball "track" in "Curvature and Creases: A Primer on Paper" and gave the properties of the "track" of the surface at one point.In this paper,the intrinsic geometric properties of the folded elements under different types of crease curves are analyzed and discussed by the mapping relationship between the properties of the developable surface and the traces of the Gaussian map,and their properties.These properties not only reflect the mathematical model of the three-dimensional folding process.The modeling method is also the basis for the different folding shape design theory of the developable surface.According to the nature of the surface crease curve,the crease curve with zero torsion is taken as the research object.The five basic curves including hyperbolic,parabolic,elliptic curve,arc curve and sine and cosine function curve are discussed as folds.Gaussian mapping of the basic expandable surface folding unit of the trace curve.The properties of the inverse Gaussian map of the basic developable surface including the cylindrical surface,the tapered surface and the tangent plane of the expandable surface on both sides of the crease curve are obtained.According to the nature of the Gaussian mapping of the surface folding unit,the forms of the traces of various possible Gaussian curves of each folding unit are established respectively,thus the database of the surface folding unit is established,and the Gaussian spherical surface of a total of 32 expandable units is established.Geometric model.And the mathematical representation corresponding to the geometric model is established.For the Gaussian mapping of the surface's folding process,the folding process of the surface is transformed into the change and analysis of the trace of the Gaussian curve.At the same time,the combination problem of multiple surfaces is also transformed into the geometric relationship of the spherical Gaussian curve.This transformation not only enables the physical model to be excessive to the geometric model,but also provides the differential geometry basis for the complex curved shape design.This paper discusses the Gaussian mapping of the basic developable surface folding model for the combination of different numbers of surface folding units,and obtains their Gaussian curve(the trace of Gaussian sphere),and analyzes the surface and straightness when combined.The characteristics of the trace of the Gaussian ball in the case where the busbar is connected as a bridge to a number of basic expandable surface folding units.The analysis method and design idea of the automatic growth of the surface element-based folding problem are realized.