Researches On Minimal Surfaces And Optimal Methods In CAD Systems

The research on minimal surface problem is one of the most active fields of interest in differential geometry.As minimal surfaces have extensive applications in engineering,it is very meaningful to introduce minimal surface into the field of CAGD(Computer Aided Geometric Design).In this thesis,some creative contributions on the topic of minimal surface modeling are given as follows.(1) Representation and construction of control mesh of minimal surfacesControl mesh is an important tool for interactive design in CAD systems.We first construct the quasi non-uniform B-spline basis and the corresponding curve and surface models in the space spanned by {sin t,cos t,sinh t,cosh t,1,t,t~2,…,t~n}. Since the knot cases are complex,we unify all the knot cases into a formula using determinate technology.Based on the new hybrid curve and surface model,the geometric construction of control mesh of the generalized helicoid is proposed. Finally,we present the control mesh representation of minimal surfaces with planar lines of curvature.These results provide an efficient tool for introducing these minimal surfaces into CAGD modeling systems,then we can produce these minimal surfaces through subdivision algorithm,and do some geometry processing with other surface in a unified way in CAGD systems.(2) Exploration and properties of parametric polynomial minimal surfaces of high degreeParametric polynomial form is the standard form of curves and surfaces in CAD systems.Based on the classical result in differential geometry,we give the sufficient conditions of harmonic parametric polynomial surfaces of degree five and six being minimal surface.Prom these conditions,several kinds of new minimal surfaces are constructed.We study the interesting geometric properties of these new minimal surfaces,such as symmetry,self-intersection and containing straight lines.We also find two new kinds of conjugate minimal surfaces,and implement the dynamic deformation between the conjugate minimal surfaces. These new minimal surfaces not only enrich the category of minimal surfaces,but also can be integrated into the current CAD systems directly.Finally,we propose three conjectures about the existence and properties of parametric polynomial minimal surfaces of arbitrary degree.(3) Approximate modeling of minimal surfaces based on linear PDEBased on the close relationship between harmonic surfaces and minimal surfaces, the properties of the harmonic B-B surface over the triangular domain are first discussed using the direction derivatives.A sufficient and necessary condition of a B-B surface over the triangular domain being a harmonic surface is obtained.We have proved that the control net of an arbitrary harmonic B-B surface over the triangular domain is fully determined by the first and second layers of control points.Then,we study the construction method of a kinds of surfaces with negative Gaussian curvature,and the result is similar with the harmonic case.In order to exchange data between PDE modeling system and CAD systems,we present a novel algorithm for approximating PDE surface by tensor product Bézier surface based on constrained optimization.We also improve this algorithm by the subdivision property of Bézier surface.(4) Solution of Plateau-Bézier problem based on squared mean curvature energyPlateau-Bézier problem is the extension of Plateau problem in Bézier form. It makes inner control points as objective solution.From the condition of mean curvature being zero,we study this problem in the case of the tensor product Bézier surfaces and the B-B surfaces over triangular domain,and give the sufficient and necessary condition that the inner control points satisfy.We also compare this method with the Dirichlet method.We can prove that,if the minimal surface with respect to the given boundary curves is parametric polynomial minimal surface with isothermal parameter,then the surface obtained by squared mean curvature method is just the parametric polynomial minimal surfaces with isothermal parameter.(5) Boundary optimization in minimal surface modelingHow to choose boundary curves to satisfy the user's requirement,is an important problem in minimal surface modeling.Firstly,based on the stretch energy, strain energy and jerk energy,we study the following problem:given partial control points of a Bézier curve,how to construct other control points such that the energy of the resulting Bézier curve is a minimum among all the energy of all Bézier curves with the same given control points.We derive the necessary and sufficient condition on the unknown control points for Bézier curves to have minimal energy.We compare the three kinds of energy-minimizing Bézier curves via curvature combs and curvature plots,and also present the collinear property of energy-minimizing quartic Bézier curves.Finally,we propose two extensions of the cubic uniform B-spline curves.First,two classes of polynomial blending functions of degree five and six are constructed.Based on the blending functions, two methods of generating piecewise polynomial curves with a shape parameter are given.It improves the approaching degree of the curves to their control polygon. Secondly,two classes of polynomial blending functions of degree three and four are presented.From these blending functions,we propose two kinds of spline curves with local shape parameter.The advantage of this method is that we can manipulate the shape of the curves locally by changing local shape parameter, while the continuity of the spline curve is unchanged.(6) Application of minimal surface in architecture designArchitecture design is the main application field of minimal surface.Based on the surface trimming technology,we first discuss the application of the new minimal surfaces and the harmonic B-B surface over triangular domain proposed in this thesis in the design of membrane structure.Finally,we propose a new concept- conical mesh of revolution,and give the simple construction method of them.We also employ them in the design of glass/steel structure.