Developable Surfaces Constructing And Approximation With Interpolation Of Two Bezier Curves

As a subset of the ruled surface, developable surface is a kind of surface that canunfold into plane without stretching or tearing, which is widely used in computer-aideddesign and manufacturing system. When the outer surface of the product are composedof the sheet metal, leather, paper or plywood, it is usually considered as a modelcomposed of multi-chip developable Bezier ruled surface; additionally, in the process ofthe side milling of machining the developable surface, it won’t produce interference. Sothe research of the developable surface is very important and meaningful. Developablesurface is mainly used in ship hulls, auto body, pipelines, airplane wings, clothing shoesand surface design. In this paper, how to design a developable surface and interpolateand approximate a surface were reviewed and the construction of developable surfaceand the approximation of a given Bezier ruled surface with developable surface wereresearched. At last, the error between the given surface and the approximation surfaceand their Gaussian curvature were analyzed.Firstly, after giving an analysis of the developable surface construct methods andshortcomings, we adopt a developable Bezier ruled surface construct method and verifythe principle. At last, based on different conditions, specific construct methods andexamples were given. The Gaussian curvature of the constructed surface and theoriginal one were calculated and analyzed to verify the constructed surface isdevelopable surface.Secondly, after the analysis of surface interpolation and approximation in recentyears, a method using piecewise developable surface to approximation a given ruledsurface was proposed. In the research process, the continuity between the adjacentsurface patches was analyzed and the continuity conditions between the adjacentsurfaces were given. The geometric continuity with the first order was also realized. Atlast, the Gaussian curvature of the approximated surface and the original one werecomputed to verify the approximation was a process to realize the developableapproximation.Finally, the approximation errors were taken into account after completing theprocess of surface approximation. In the paper, we first defined the approximationerrors and then presented a method to achieve a surface approximation with consideringthe surface approximation errors.