The Construction,Interpolation And Shape Optimization Of The Generalized H-Bezier Developable Surface

The developable surface is a kind of special ruled surface,which can be unfolded into a complete plane under the condition of no stretching,tearing or folding.Owing to this characteristic,the developable surface has extensive application value on some manufacturing fields who use nonretractable materials.Therefore,the researches on how to construct and optimize developable surface are always significant in CAD/CAM field.Although the developable surface designed by traditional Bezier model hold simple structure and is easy to realized,it also exists some shortcomings of being difficult to adjust and optimize its shape.Since H-Bezier model can modify and optimize its shape by introducing shape parameters,the study of developable surface based on H-Bezier model has important meaning in theory and application.The construction,interpolation and shape optimization of a class of generalized cubic H-Bezier developable surface with shape parameters are studied in this paper.The specific work and research results are shown below:(1)A class of generalized cubic H-Bezier basis functions with three local shape parameters and its properties are proposed at first.And then the generalized cubic H-Bezier curves and surfaces with shape parameters are presented;we analyze the influence rules of shape parameters on the presented curves and surfaces,and discuss the precise representation of hyperbolae and catenaries with generalized cubic H-Bezier curves.Finally,the G1,G2 and G1 continuity conditions of generalized cubic H-Bézier curves and surfaces are derived,respectively;meanwhile,the concrete algorithms are given.The numerical examples show that the generalized cubic H-Bézier curves and surfaces can be used as a powerful supplement of complex curves and surfaces design in CAD/CAM system.(2)Based on the duality principle between point and plane in 3D projective space,a class of control planes with H-Bezier basis is generated by utilizing the generalized cubic H-Bézier basis.We construct the generalized cubic H-Bezier enveloping and spine curve developable surfaces in terms of the generated control planes,and give their explicit expressions.Combining theoretical analysis with numerical examples,we study the properties of developable surfaces and the geometrical significance of shape parameters.Furthermore,we deduce the continuity conditions for G1 continuity,Farin-Boehm G2 continuity and G2 Beta continuity between two cubic generalized H-Bezier developable surfaces,and analyze the influence laws of shape parameters and scale parameter on the shape of composite developable surface.The modelling examples show that the generalized H-Bezier developable surface is not only easy to computing,but also can be convenient to adjust its local or global shape;in addition,the introduction of shape parameters provides more degrees of freedom for shape optimization of developable surface.(3)Geodesic and line of curvature are two special curves on engineering surface,they are not only closely related to developable surface,but also play an important role in geometric design and surface analysis.Firstly,we construct a local controlled developable H-Bezier surface possessing a given generalized cubic H-Bezier curve as its line of curvature or geodesic,and give the necessary and sufficient conditions for the constructed developable surface to be a cylinder or a cone.Secondly,the influence rules of shape parameters on the shape of developable surface are discussed in detail,and the G1 continuity conditions for two adjacent interpolating developable H-Bezier surfaces are deduced.Finally,we give some representative numerical examples to verify the simplicity and validity of the proposed method.(4)The shape optimization design of developable surface is a key and tough technology in CAD/CAM field.In this paper,we research the shape optimization of the generalized cubic H-Bezier developable surface by using the cuckoo search(CS)algorithm.Firstly,we transform the shape optimization problem of developable surface into minimization problem of arc length,energy and rate of curvature of its dual curve in terms of duality principle,and establish three corresponding optimization models;and then using the CS algorithm to solve these models.Moreover,the sensitivity for two key parameters of population size and probability of discovery are analyzed.Experiment results show that the objective functions are not sensitive to the two key parameters in CS algorithms,which proves the effectiveness of proposed method further.