The Modeling Of Interpolation Of Free Form Curve Networks By Approximately Developable Patches

When developing a developable surface into a plane，there is no tensile andtorsional deformation happened. Because of this characteristic, make thedevelopable surface to be a wide range of applications in the engineering field, andcaused extensive research in the area of CAGD(Computer Aided Geometric Design).In engineering applications, we often met the conditions that given the mesh surface,and interpolate the mesh grid with approximate developable surface patch. Thisarticle is abstracted as the problem of the modeling of approximate developablesurface to interpolate the freeform curve networks. Use approximate developablesurface to interpolate the curve networks, and make the adjacent surface patchessmooth connected.The main contents of the paper:Select the appropriate interpolation patch. Analysis the characteristics ofdifferent interpolation patch, choose the patch can not only optimize the Gaussiancurvature, but also ensure the adjacent patches smooth connected. The boundarydifferential characteristics of Gregory patch are the same with the Bezier patch withthe same boundary, it can ensure the smooth connection of the patches through thecontrol points related to the first order differential component of the boundary, canalso optimize the developability the patch through the internal control points. Thisarticle use the Gregory patch to construct the approximate developable surface.The Gaussian curvature at the corner of the Gregory patch is only relevant tothe control points related to the first order differential component. When the firstorder continuity of adjacent surfaces is confirmed, the Gaussian curvature isdetermined value, and can’t be optimized later. The method to solve the problem inthe article is, to make the constraints of Gaussian curvature when determine theconditions of the first order continuity between the adjacent patches.Simplify the objective function of the developability optimization. Simplify theobjective function of continuous integral form into the sum of discrete values, andreduce the difficulty of calculate in the process of optimization. Initialize the valueof some part of internal unknown control points, reducing the number of unknownvalues, and improve the computational efficiency.Use the genetic algorithm to optimize the developability of the surface. Forcomplex objective function, the traditional optimization algorithms are difficult tocalculate, so this article use the genetic algorithm to optimize the developability ofthe surface, which is more mature in technology in the intelligent optimization algorithm. And finished the example of interpolate the given boundary with theapproximate developable surface. The developability of the Gaussian curvatureoptimized surface is greatly improved than the initial interpolation surface.