Research On Geometric Modeling Methods For Complex Minimal Surfaces

The minimal surface is a special surface with important application in geometric modeling and design.The application minimal surface range is very wide,in the architectural design not only has good ornamental,but also saves building materials.In modern manufacturing industry,such as aircraft and ships,aircraft manufacturing application cannot do without even the minimal surface,and geometric modeling of minimal surfaces has been an important issue in the research field of the computer aided design and computer graphics.In order to further improve the modeling method of complex minimal surface,this paper deeply studies the modeling method of minimal surface,and puts forward a new method of modeling.From the approximate exact representation of arbitrary degree and complex boundary representation of the two angles of geometric modeling method for complex minimal surfaces is studied,mainly including three aspects.1)Explicit form of parametric polynomial minimal surfaces with arbitrary degree.As we know that CAD system is the core of the geometric modeling technology.From this viewpoint,we propose an explicit parametric form of a class of parametric polynomial minimal surfaces with arbitrary degree.The proposed new minimal surface has some fascinated properties such as symmetry,containing straight lines and self-intersections.Based on shape characteristics,the proposed minimal surface can be classified into four categories with respect to n = 4k-1,n = 4k;n = 4k + 1 and n =4k + 2,where n is the degree of the minimal surface and k is a positive integral number.At last,the explicit parametric form of the conjugate minimal surface is also given,and its isometric deformation is realized.2)High quality quadrilateral mesh generation method for complex boundary minimal surface modeling.High-quality quadrilateral mesh generation is an important and challenging problem in the field of Computer Aided-Design,Isogeometric Analysis and Computer Graphics.In this paper,we propose a new framework for High quality quadrilateral mesh generation method for complex boundary minimal surface modeling.Compared with existing methods,the proposed method can generate high-quality quad mesh with fewer extraordinary vertices and nice mesh qualities,such as smoothness,uniformity and orthogonality,as shown in the presented modeling examples.3)Constructing Discrete Minimal Surfaces with Quadrilateral Meshes from Described Boundary.Efficient modeling of minimal surfaces is a challenging problem and hot topic in the field of geometric design and computation.Given a boundary closed polylines,research on how to construct the discrete minimal surface of quadrilateral mesh with its boundary.Firstly,the mathematical definition of discerete minimal surface with quadrilateral mesh is given from the intrinsic differential-geometry metric of surfaces.Based on the length-preserving boundary projection method,quadrilateral mesh generation,radial basis function interpolation mapping and nonlinear optimization techniques,a novel framework is proposed to construct discrete minimal surfaces with quadrilateral meshes from a described boundary closed discrete polylines.Finally,the effectiveness of the proposed method is verified through a number of modeling examples.The method can achieve high quality modeling of minimal surface of quadrilateral mesh,and it has a certain application value in the field of building geometry.