Research On Developable Surface In Curve And Surface Modeling

Developable surface,as a kind of special surface with Gaussian curvature equal to 0,can be expanded on a plane without tearing and stretching.Developable surface construction is often involved in aircraft,ship,automobile and other industrial production,clothing and shoes,architecture.Because of its wide range of applications,the construction method of developable surface has always been a hot issue discussed by scholars in the field of Computer Aided Geometric Design,in which the methed of point face duality and surface inverse construction method with certain geometric constraints are the mainstream construction methods of developable surface.In recent years,the research on developable surface emerges one after another,but there are still some deficiencies.This paper makes a further exploration on these two methods.The property of Poisson basis function is similar to the Bezier curve,and the parameter definition domain of Poisson basis function is[0,+?),which is larger than that of Bezier basis function.In the third chapter,based on the duality principle of points and planes in projective space,the family of single-parameter control planes represented by Poisson basis functions is calculated by Pliicker coordinate theorem to obtain the expression of developable Poisson surfaces.This method avoids a large number of calculations of characteristic equations and makes the design more direct and effective.At the same time,the definition domain of the parameters of Poisson basis function is larger,which makes the adjustable range of developable Poisson surface modeling larger.In the fourth chapter of this thesis,developable surface pencil pairs with Natural pair and Conjugate pair(hereinafter referred to as special pairs)as common asymptotes are proposed and further applied to surface modeling through examples.By establishing the Frenet frame of special pairs in three-dimensional Euclidean space,we express surface pencil pairs as a linear combination of Frenet frame,and derive the necessary and sufficient conditions of surface pencil pairs with special pairs as common asymptotes.Furthermore,the surface pencil pairs are extended to developable surface pencil pairs,as the selection of marching-scale functions are different,developable surface pencil pairs have good adjustability.Finally,when spatial curve is a cylindrical helix or a slant helix,examples of developable surface pencil pairs with special pairs as common asymptotes are given,respectively.