The Structure And Analysis Of Developable Surface Based On The Curve And Surface

The developable surfaces are special ruled surfaces whose Gaussian curvature vanish at all surface points. They have many important properties. For example, developable surfaces can be developed into a plane without stretching and tearing, and they are envelopes of one parameter sets of planes and so on. Because of these properties, they have great application value in the surface modeling. For example, if the surface of an object is developable in the design of the entity appearance, people can design it in a plane.In the texture mapping tech-nology of Computer Graphics, a planar picture can be mapped on the developable surfaces without stretching and tearing. So, that how to construct the developable surfaces has become an important problem which should be solved currently according to actual engineering re-quirements.Therefore, based on the known geometry condition of the curve and surface, further re-search and discussion, which mainly focus on how to construct the developable surface as well as relative issues, has been put forward.Firstly, through the general theory and method being perfected that the developable sur-face can be constructed by means of the surface and curve, the expression form of developable surface is obtained and the developable surface is classified. Based on establishing the map-ping relationship between two surfaces, the paper realizes the mapping analysis of the whole and partial between them, accurately grasps the deformation conditions of surface geometry elements, and verifies the theory and method by means of example.Secondly, the paper systematically puts forward theory and methods that how the deve-lopable tangent surface of revolution surface is constructed and the mapping is analysed be-tween the developable tangent surface and revolution surface, meanwhile establishes the ma-thematical model of the developable tangent cylinder and the developable tangent cone of rev-olution surface as well as the mapping relationship between them. According to the theoretical analysis of the differential length ratio of revolution surface and developable tangent surface, the differential equations of maximum deformation curve and equidistant curve are derived in the mappings. Based on the deformation analysis of the whole and partial, the deformation conditions can be accurately grasped in the mapping relationship between revolution surface and its developable tangent surface.Thirdly, on the basis of the study about developable tangent surface, the theory and me-thod that how to construct developable surface are obtained based on the geometry condition of the surface and curve, including constructing the developable surface by means of a curve and known tangent surface or two curves, and the expression form of developable surface can finally been established.Fourthly, as the application of theory and method mentioned above, three examples of application such as surface mapping, approximate expansion of unextensible surface and the design of developable surface are given based on the construction and analysis of developable surface.